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2026-01-01
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2026-02-28
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<p>154 Learners</p>
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<p>184 Learners</p>
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<p>Last updated on<strong>October 21, 2025</strong></p>
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<p>Last updated on<strong>October 21, 2025</strong></p>
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<p>A vector is a quantity that has both magnitude and direction. It is represented using an arrow. The length of the arrow reflects the magnitude, and the orientation indicates the direction.</p>
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<p>A vector is a quantity that has both magnitude and direction. It is represented using an arrow. The length of the arrow reflects the magnitude, and the orientation indicates the direction.</p>
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<h2>What are Vector Equations?</h2>
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<h2>What are Vector Equations?</h2>
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<p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>
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<p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>
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<p>▶</p>
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<p>▶</p>
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<p>Vector equations use direction vectors and<a>variables</a>to describe planes or lines in three-dimensional space. A vector with a<a>magnitude</a><a>of</a>1 is a unit vector. In a three-dimensional space, all positions are described using 3 axes, the x-axis, y-axis, and z-axis, and each axis has a unit vector. The unit vector for the x-axis is \(\hat {i}\), the unit vector for the y-axis is \(\hat {j}\), and the unit vector for the z-axis is \(\hat {k}\).</p>
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<p>Vector equations use direction vectors and<a>variables</a>to describe planes or lines in three-dimensional space. A vector with a<a>magnitude</a><a>of</a>1 is a unit vector. In a three-dimensional space, all positions are described using 3 axes, the x-axis, y-axis, and z-axis, and each axis has a unit vector. The unit vector for the x-axis is \(\hat {i}\), the unit vector for the y-axis is \(\hat {j}\), and the unit vector for the z-axis is \(\hat {k}\).</p>
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<p>In a 3D space, a vector is written as: \(\vec{r} = {x {\hat {i}}} + {y {\hat {j}}} + {z {\hat {k}}}\)Where x, y, and z represent the scalar components of the vector. The vector<a>equation</a>of a line in three-dimensional space is: \(\vec{r} = \vec{a} + \lambda{{\vec{b}} }\) where \(λ∈R\) The vector equation of a plane in three-dimensional space is: \(\vec{r} \cdot {\vec n} = d\). </p>
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<p>In a 3D space, a vector is written as: \(\vec{r} = {x {\hat {i}}} + {y {\hat {j}}} + {z {\hat {k}}}\)Where x, y, and z represent the scalar components of the vector. The vector<a>equation</a>of a line in three-dimensional space is: \(\vec{r} = \vec{a} + \lambda{{\vec{b}} }\) where \(λ∈R\) The vector equation of a plane in three-dimensional space is: \(\vec{r} \cdot {\vec n} = d\). </p>
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<h2>Vector Equations vs Cartesian Equations</h2>
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<h2>Vector Equations vs Cartesian Equations</h2>
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<p>A Cartesian equation is an<a>algebraic expression</a>that shows the relationship between variables (typically x, y, and z) using the Cartesian coordinate system. It is used to describe<a>geometric</a>figures such as lines, curves, and surfaces in<a>terms</a>of their positions on a coordinate plane or in space. Here are some key differences between vector and Cartesian equations. </p>
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<p>A Cartesian equation is an<a>algebraic expression</a>that shows the relationship between variables (typically x, y, and z) using the Cartesian coordinate system. It is used to describe<a>geometric</a>figures such as lines, curves, and surfaces in<a>terms</a>of their positions on a coordinate plane or in space. Here are some key differences between vector and Cartesian equations. </p>
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<p><strong>Feature</strong></p>
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<p><strong>Feature</strong></p>
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<p><strong>Vector Equation</strong></p>
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<p><strong>Vector Equation</strong></p>
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<p><strong>Cartesian Equation</strong></p>
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<p><strong>Cartesian Equation</strong></p>
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<p><strong>Form</strong></p>
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<p><strong>Form</strong></p>
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<p>Uses vectors and vector units.</p>
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<p>Uses vectors and vector units.</p>
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<p>Uses x, y, and z coordinates directly.</p>
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<p>Uses x, y, and z coordinates directly.</p>
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<p><strong>Representation</strong></p>
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<p><strong>Representation</strong></p>
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Line : \({\vec r = \vec a + \lambda \vec b}\) Plane: \(\vec r \cdot \hat n = d\) Line: \({{x - x_1} \over a} = {{y - y_1} \over b} = {{z-z_1} \over c} \) Plane: \(ax + by + cz = d\)<p><strong>Nature</strong></p>
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Line : \({\vec r = \vec a + \lambda \vec b}\) Plane: \(\vec r \cdot \hat n = d\) Line: \({{x - x_1} \over a} = {{y - y_1} \over b} = {{z-z_1} \over c} \) Plane: \(ax + by + cz = d\)<p><strong>Nature</strong></p>
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<p>Vector-based, compact, and geometric</p>
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<p>Vector-based, compact, and geometric</p>
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<p>coordinate-based, algebraic</p>
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<p>coordinate-based, algebraic</p>
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<p><strong>Variables used</strong></p>
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<p><strong>Variables used</strong></p>
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<p>Vectors and scalars</p>
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<p>Vectors and scalars</p>
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Coordinates<p><strong>Visualization</strong></p>
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Coordinates<p><strong>Visualization</strong></p>
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Easier to visualize in space using direction and position vectors<p>Useful for calculations and plotting on a coordinate grid</p>
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Easier to visualize in space using direction and position vectors<p>Useful for calculations and plotting on a coordinate grid</p>
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<p><strong>Used in</strong></p>
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<p><strong>Used in</strong></p>
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Physics and mechanics<p>Algebraic calculations and<a>graphing</a></p>
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Physics and mechanics<p>Algebraic calculations and<a>graphing</a></p>
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<h2>What is the Vector Equation of a Line?</h2>
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<h2>What is the Vector Equation of a Line?</h2>
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<p>In a three-dimensional space, the vector equation of a line uses vectors to describe all the points that lie on a straight line. The vector equation of a line that passes through a single point: \(\vec r = \vec a + \lambda \vec b \) Here, \(\vec r\) represents the position vector of any point on the line While \(\vec a\) represents the position vector of a fixed point through which the line passes \(\vec b\) = direction vector of the line and \(\lambda\) = scalar parameter that varies.</p>
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<p>In a three-dimensional space, the vector equation of a line uses vectors to describe all the points that lie on a straight line. The vector equation of a line that passes through a single point: \(\vec r = \vec a + \lambda \vec b \) Here, \(\vec r\) represents the position vector of any point on the line While \(\vec a\) represents the position vector of a fixed point through which the line passes \(\vec b\) = direction vector of the line and \(\lambda\) = scalar parameter that varies.</p>
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<p>Vector equation for a line passing through two points: \(\vec r = \vec a + \lambda(\vec b - \vec a)\)</p>
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<p>Vector equation for a line passing through two points: \(\vec r = \vec a + \lambda(\vec b - \vec a)\)</p>
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<p> Here, \(\vec a\) (position vector of point A) \(\vec a = x_1 \hat i + y_1 \hat j + z_1 \hat k \) \(\vec b\) (position vector of point B) \(\vec b = x_2 \hat i + y_2 \hat j + z_2 \hat k \)</p>
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<p> Here, \(\vec a\) (position vector of point A) \(\vec a = x_1 \hat i + y_1 \hat j + z_1 \hat k \) \(\vec b\) (position vector of point B) \(\vec b = x_2 \hat i + y_2 \hat j + z_2 \hat k \)</p>
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<p>The direction vector is \(\vec b - \vec a\), pointing from point A to point B. \(λ∈R\) is a scalar parameter </p>
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<p>The direction vector is \(\vec b - \vec a\), pointing from point A to point B. \(λ∈R\) is a scalar parameter </p>
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<h3>Explore Our Programs</h3>
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<h3>Explore Our Programs</h3>
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<h2>What are Vector Equations of a Plane?</h2>
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<h2>What are Vector Equations of a Plane?</h2>
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<p>The vector equation represents motion from a fixed point in a plane along two independent directions. \(\vec r = \vec a + s \vec u + t \vec v\)</p>
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<p>The vector equation represents motion from a fixed point in a plane along two independent directions. \(\vec r = \vec a + s \vec u + t \vec v\)</p>
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<p>Here, \(\vec r\) is the position vector of any point P(x, y, z) lying on the plane \(\vec a\) is vector of a specific point on the plane \(\vec u, \vec v\) are two non-parallel direction vectors that lie on the plane. s and t are scalar parameters; they can be any<a></a><a>real number</a>.</p>
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<p>Here, \(\vec r\) is the position vector of any point P(x, y, z) lying on the plane \(\vec a\) is vector of a specific point on the plane \(\vec u, \vec v\) are two non-parallel direction vectors that lie on the plane. s and t are scalar parameters; they can be any<a></a><a>real number</a>.</p>
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<h2>Tips and Tricks to Master Vector Equations</h2>
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<h2>Tips and Tricks to Master Vector Equations</h2>
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<p>Vector equations help us represent lines, planes, and directions in space using vectors. In this section, we will discuss some tips and tricks to master vector equations. This help students to solve problems in physics and mathematics much more easily.</p>
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<p>Vector equations help us represent lines, planes, and directions in space using vectors. In this section, we will discuss some tips and tricks to master vector equations. This help students to solve problems in physics and mathematics much more easily.</p>
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<ul><li><p>Always separate the components, that is, break the vectors into x, y, and z components before solving. </p>
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<ul><li><p>Always separate the components, that is, break the vectors into x, y, and z components before solving. </p>
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</li>
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</li>
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<li><p>Memorize the basic<a>formulas</a>like: Angle between vectors: \(cos{\theta} = {{{\vec a \cdot \vec b} \over |\vec a| |\vec b|}}\). Perpendicular vectors: Dot<a>product</a>= 0 \({(\vec a \cdot \vec b = 0 )}\)Magnitude: \({{|\vec v| = \sqrt {x^2 + y^2 + z^2}}}\)</p>
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<li><p>Memorize the basic<a>formulas</a>like: Angle between vectors: \(cos{\theta} = {{{\vec a \cdot \vec b} \over |\vec a| |\vec b|}}\). Perpendicular vectors: Dot<a>product</a>= 0 \({(\vec a \cdot \vec b = 0 )}\)Magnitude: \({{|\vec v| = \sqrt {x^2 + y^2 + z^2}}}\)</p>
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</li>
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</li>
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<li><p>If a vector is a multiple of another vector, then the vectors are parallel. For example, (2, 4) is parallel to (1, 2).</p>
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<li><p>If a vector is a multiple of another vector, then the vectors are parallel. For example, (2, 4) is parallel to (1, 2).</p>
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</li>
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</li>
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<li><p>Students can draw a simple diagram to visualize how the vectors look, where they point, and how they relate to each other. This help students to understand if they are parallel, perpendicular, or intersecting.</p>
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<li><p>Students can draw a simple diagram to visualize how the vectors look, where they point, and how they relate to each other. This help students to understand if they are parallel, perpendicular, or intersecting.</p>
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</li>
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</li>
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</ul><h2>Common Mistakes and How to Avoid Them in Vector Equations</h2>
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</ul><h2>Common Mistakes and How to Avoid Them in Vector Equations</h2>
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<p>Vector equations are crucial in solving 3D geometry problems and various subjects like physics and engineering. To ensure a better understanding and precise results, avoid the following frequent errors that get overlooked:</p>
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<p>Vector equations are crucial in solving 3D geometry problems and various subjects like physics and engineering. To ensure a better understanding and precise results, avoid the following frequent errors that get overlooked:</p>
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<h2>Real-Life Applications of Vector Equations</h2>
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<h2>Real-Life Applications of Vector Equations</h2>
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<p>Vector equations represent quantities having both magnitude and direction. They play a crucial role in various real-life, direction-based calculations:</p>
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<p>Vector equations represent quantities having both magnitude and direction. They play a crucial role in various real-life, direction-based calculations:</p>
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<ul><li><strong>Force and motion analysis</strong>: Vector equations describe the position, velocity, and acceleration of moving objects in space. This helps analyze trajectories, forces, and dynamics in mechanics.</li>
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<ul><li><strong>Force and motion analysis</strong>: Vector equations describe the position, velocity, and acceleration of moving objects in space. This helps analyze trajectories, forces, and dynamics in mechanics.</li>
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<li><strong>Graphics and Animations</strong>: Vectors represent object positions, directions, and transformations in 2D or 3D space. They are used in calculations for moving graphic objects and rendering in games or simulations.</li>
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<li><strong>Graphics and Animations</strong>: Vectors represent object positions, directions, and transformations in 2D or 3D space. They are used in calculations for moving graphic objects and rendering in games or simulations.</li>
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</ul><p> \(\vec p_{new} = \vec p _{original} + \vec v \cdot t \) </p>
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</ul><p> \(\vec p_{new} = \vec p _{original} + \vec v \cdot t \) </p>
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<ul><li><strong>GPS and Navigation Systems</strong>: GPS uses vector<a>math</a>to compute displacement, direction, and distance between two points on Earth, determining shortest paths.</li>
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<ul><li><strong>GPS and Navigation Systems</strong>: GPS uses vector<a>math</a>to compute displacement, direction, and distance between two points on Earth, determining shortest paths.</li>
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</ul><ul><li><strong>Robotics: </strong>Vector equations are used in machinery like robotic arms to reach a point in 3D space. Movement and orientation are modeled using vector equations for precise positioning. </li>
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</ul><ul><li><strong>Robotics: </strong>Vector equations are used in machinery like robotic arms to reach a point in 3D space. Movement and orientation are modeled using vector equations for precise positioning. </li>
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<li><strong>Engineering- structural analysis</strong>: Engineers use vectors to model forces acting on structures such as bridges and buildings. Vector equations help in calculating stress, strain, and stability.</li>
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<li><strong>Engineering- structural analysis</strong>: Engineers use vectors to model forces acting on structures such as bridges and buildings. Vector equations help in calculating stress, strain, and stability.</li>
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</ul><h3>Problem 1</h3>
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</ul><h3>Problem 1</h3>
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<p>Find the vector equation of a line passing through a point A(1,2) and having a direction vector d = 34</p>
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<p>Find the vector equation of a line passing through a point A(1,2) and having a direction vector d = 34</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>\(\vec r (\lambda) = \begin{bmatrix} 1 + 3\lambda \\ 2 + 4\lambda \end{bmatrix} \)</p>
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<p>\(\vec r (\lambda) = \begin{bmatrix} 1 + 3\lambda \\ 2 + 4\lambda \end{bmatrix} \)</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>The vector equation of a line is: \(\vec r (\lambda) = \vec a + \lambda \vec d\) Where: \(\vec a\) = \(\begin{bmatrix} 1 \\ 2 \end{bmatrix} \) position vector of A \(\vec d = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \) \(\vec r (\lambda) = \begin{bmatrix} 1 \\ 2 \end{bmatrix} + \lambda\begin{bmatrix} 4 \\ 3 \end{bmatrix} \) \(= \begin{bmatrix} 1 + 3\lambda \\ 2 + 4\lambda \end{bmatrix} \)</p>
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<p>The vector equation of a line is: \(\vec r (\lambda) = \vec a + \lambda \vec d\) Where: \(\vec a\) = \(\begin{bmatrix} 1 \\ 2 \end{bmatrix} \) position vector of A \(\vec d = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \) \(\vec r (\lambda) = \begin{bmatrix} 1 \\ 2 \end{bmatrix} + \lambda\begin{bmatrix} 4 \\ 3 \end{bmatrix} \) \(= \begin{bmatrix} 1 + 3\lambda \\ 2 + 4\lambda \end{bmatrix} \)</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 2</h3>
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<h3>Problem 2</h3>
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<p>Let u = [2, -1, 4] and v = [-3, 0, 1]. Find u + v</p>
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<p>Let u = [2, -1, 4] and v = [-3, 0, 1]. Find u + v</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>\(\vec u + \vec v = {{\begin{bmatrix} -1 \\ -1 \\ 5 \\ \end{bmatrix} }}\)</p>
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<p>\(\vec u + \vec v = {{\begin{bmatrix} -1 \\ -1 \\ 5 \\ \end{bmatrix} }}\)</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>\(\vec u + \vec v = \begin{bmatrix} 2 \\ -1 \\ 4 \\ \end{bmatrix} + \begin{bmatrix} -3 \\ 0\\ 1\\ \end{bmatrix} \)</p>
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<p>\(\vec u + \vec v = \begin{bmatrix} 2 \\ -1 \\ 4 \\ \end{bmatrix} + \begin{bmatrix} -3 \\ 0\\ 1\\ \end{bmatrix} \)</p>
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<p>\(\begin{bmatrix} 2 - 3 \\ -1 + 0\\ 4 + 1 \\ \end{bmatrix} \)</p>
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<p>\(\begin{bmatrix} 2 - 3 \\ -1 + 0\\ 4 + 1 \\ \end{bmatrix} \)</p>
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<p>\(\begin{bmatrix} -1 \\ -1\\ 5 \\ \end{bmatrix} \)</p>
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<p>\(\begin{bmatrix} -1 \\ -1\\ 5 \\ \end{bmatrix} \)</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 3</h3>
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<h3>Problem 3</h3>
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<p>Let a = [3 4] and b = [4 3]. Find the angle Θ between them</p>
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<p>Let a = [3 4] and b = [4 3]. Find the angle Θ between them</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>\(\theta = 16.26 ^\circ\)</p>
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<p>\(\theta = 16.26 ^\circ\)</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Finding \(\vec a \cdot \vec b\) to find the angle \({{{\theta }}}\),</p>
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<p>Finding \(\vec a \cdot \vec b\) to find the angle \({{{\theta }}}\),</p>
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<p>\({{\vec a \cdot \vec b }} = {3 \times 4} + {4 \times 3}\\ = 12 + 12 \\ = 24\)</p>
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<p>\({{\vec a \cdot \vec b }} = {3 \times 4} + {4 \times 3}\\ = 12 + 12 \\ = 24\)</p>
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<p>Finding the magnitudes of \(\vec a {\text { and }} \vec b\) \(|\vec a| = \sqrt {3^2 + 4^2} = 5,\\ |\vec b| = \sqrt {4^2 + 3^2} = 5\\ \)</p>
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<p>Finding the magnitudes of \(\vec a {\text { and }} \vec b\) \(|\vec a| = \sqrt {3^2 + 4^2} = 5,\\ |\vec b| = \sqrt {4^2 + 3^2} = 5\\ \)</p>
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<p>To find the value of \({{{\theta }}}\), we use the formula: </p>
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<p>To find the value of \({{{\theta }}}\), we use the formula: </p>
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<p>\({{\vec a \cdot \vec b }} = {{|\vec a| |\vec b| cos {\theta}}}\)</p>
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<p>\({{\vec a \cdot \vec b }} = {{|\vec a| |\vec b| cos {\theta}}}\)</p>
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<p> \({{\vec a \cdot \vec b }} = {{|\vec a| |\vec b| cos {\theta}}} \implies 24= 5 \times 5 cos {\theta} \\\implies cos {\theta } = {24 \over 25}\\ \implies {\theta } = cos^{-1} {{({24\over 25})}} \approx 16.26 ^\circ\)</p>
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<p> \({{\vec a \cdot \vec b }} = {{|\vec a| |\vec b| cos {\theta}}} \implies 24= 5 \times 5 cos {\theta} \\\implies cos {\theta } = {24 \over 25}\\ \implies {\theta } = cos^{-1} {{({24\over 25})}} \approx 16.26 ^\circ\)</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 4</h3>
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<h3>Problem 4</h3>
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<p>Project vector a = [2 4] onto vector b = [1 2]</p>
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<p>Project vector a = [2 4] onto vector b = [1 2]</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>\({{{\text { Proj} _ {\vec b} {\vec a}}}} = \begin{bmatrix} 2 \\ 4 \\ \end {bmatrix} \)</p>
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<p>\({{{\text { Proj} _ {\vec b} {\vec a}}}} = \begin{bmatrix} 2 \\ 4 \\ \end {bmatrix} \)</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>\( \text{Proj}_{\vec{b}} \vec{a} = {({\vec a \cdot \vec b \over \vec b \cdot \vec b})} \vec b \)</p>
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<p>\( \text{Proj}_{\vec{b}} \vec{a} = {({\vec a \cdot \vec b \over \vec b \cdot \vec b})} \vec b \)</p>
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<p>Finding \({\vec a \cdot \vec b} = {2 \times 1 + 4 \times 2} = 10\\ {\vec b \cdot \vec b} = {1^2 + 2^2} = 5\\ \)</p>
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<p>Finding \({\vec a \cdot \vec b} = {2 \times 1 + 4 \times 2} = 10\\ {\vec b \cdot \vec b} = {1^2 + 2^2} = 5\\ \)</p>
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<p>\( \text{Proj}_{\vec{b}} \vec{a} = {{({10 \over 5})}} \begin{bmatrix}1 \\ 2\\ \end {bmatrix}\\ = 2 \begin{bmatrix}1 \\ 2\\ \end {bmatrix} \\ = \begin{bmatrix}2 \\ 4\\ \end {bmatrix}\)</p>
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<p>\( \text{Proj}_{\vec{b}} \vec{a} = {{({10 \over 5})}} \begin{bmatrix}1 \\ 2\\ \end {bmatrix}\\ = 2 \begin{bmatrix}1 \\ 2\\ \end {bmatrix} \\ = \begin{bmatrix}2 \\ 4\\ \end {bmatrix}\)</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 5</h3>
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<h3>Problem 5</h3>
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<p>Find the vector equation of a plane passing through point A(1,0,2) with direction vectors</p>
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<p>Find the vector equation of a plane passing through point A(1,0,2) with direction vectors</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>\(\vec r (s, t) = \begin {bmatrix} 1 + s \\ 2s + t\\ 2 + 3t\\ \end {bmatrix}\)</p>
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<p>\(\vec r (s, t) = \begin {bmatrix} 1 + s \\ 2s + t\\ 2 + 3t\\ \end {bmatrix}\)</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>The vector equation for the palne is: </p>
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<p>The vector equation for the palne is: </p>
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<p>\(\vec r (s, t) = \vec a + s \vec u + t \vec v\)</p>
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<p>\(\vec r (s, t) = \vec a + s \vec u + t \vec v\)</p>
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<p>Here, a = (1, 0, 2)</p>
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<p>Here, a = (1, 0, 2)</p>
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<p>\(\vec u = \begin {bmatrix} 1\\ 2\\ 0\\ \end {bmatrix}\) \(\vec v = \begin {bmatrix} 0\\ 1\\ 3\\ \end {bmatrix}\)</p>
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<p>\(\vec u = \begin {bmatrix} 1\\ 2\\ 0\\ \end {bmatrix}\) \(\vec v = \begin {bmatrix} 0\\ 1\\ 3\\ \end {bmatrix}\)</p>
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<p>So, \(\vec r (s, t) = \begin {bmatrix} 1\\ 0\\ 2\\ \end {bmatrix} + s \begin {bmatrix} 1\\ 2\\ 0\\ \end {bmatrix} + t \begin {bmatrix} 0\\ 1\\ 3\\ \end {bmatrix}\)</p>
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<p>So, \(\vec r (s, t) = \begin {bmatrix} 1\\ 0\\ 2\\ \end {bmatrix} + s \begin {bmatrix} 1\\ 2\\ 0\\ \end {bmatrix} + t \begin {bmatrix} 0\\ 1\\ 3\\ \end {bmatrix}\)</p>
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<p>\(= \begin {bmatrix} 1 + s\\ 2s + t \\ 2s + 3t\\ \end {bmatrix}\)</p>
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<p>\(= \begin {bmatrix} 1 + s\\ 2s + t \\ 2s + 3t\\ \end {bmatrix}\)</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h2>FAQs on Vector Equations</h2>
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<h2>FAQs on Vector Equations</h2>
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<h3>1.How to find a vector?</h3>
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<h3>1.How to find a vector?</h3>
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<p>A vector is found by subtracting the coordinates of two points. On the other hand, if we know the magnitude and direction, multiplying the magnitude by a unit vector in the given direction gives us the vector. </p>
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<p>A vector is found by subtracting the coordinates of two points. On the other hand, if we know the magnitude and direction, multiplying the magnitude by a unit vector in the given direction gives us the vector. </p>
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<h3>2.How to find symmetric equations?</h3>
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<h3>2.How to find symmetric equations?</h3>
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<p>To find symmetric equations of a line in 3D, start with its vector or parametric form </p>
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<p>To find symmetric equations of a line in 3D, start with its vector or parametric form </p>
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<p> Eliminate parameter t: x-x0a=y-y0b=z-z0c </p>
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<p> Eliminate parameter t: x-x0a=y-y0b=z-z0c </p>
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<p> We assume that a, b, and c are not zero. So, this is the symmetric form of the line.</p>
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<p> We assume that a, b, and c are not zero. So, this is the symmetric form of the line.</p>
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<h3>3.Parametric vs symmetric equations</h3>
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<h3>3.Parametric vs symmetric equations</h3>
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<p>Parametric Equations</p>
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<p>Parametric Equations</p>
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<p>Symmetric equations</p>
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<p>Symmetric equations</p>
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<p>Step-by-step movement along the line</p>
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<p>Step-by-step movement along the line</p>
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<p>full line describing coordinate dependency without a parameter.</p>
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<p>full line describing coordinate dependency without a parameter.</p>
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<p>Parameters, like t, are involved </p>
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<p>Parameters, like t, are involved </p>
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<p>No parameter, all variables are related directly</p>
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<p>No parameter, all variables are related directly</p>
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<p>represents the position of a specific point on a line</p>
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<p>represents the position of a specific point on a line</p>
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<p>Represents the full line in a compact form</p>
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<p>Represents the full line in a compact form</p>
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<p>Derived from a point and a direction vector</p>
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<p>Derived from a point and a direction vector</p>
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<p>Derived from eliminating t from parametric equations</p>
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<p>Derived from eliminating t from parametric equations</p>
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<p>Requires parameter substitution for use</p>
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<p>Requires parameter substitution for use</p>
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<p>Symmetric equations are not required if any direction vector component is 0 (<a>division by zero</a>)</p>
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<p>Symmetric equations are not required if any direction vector component is 0 (<a>division by zero</a>)</p>
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<p>Good for motion/path modeling</p>
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<p>Good for motion/path modeling</p>
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<p>Good for algebraic manipulation or solving systems</p>
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<p>Good for algebraic manipulation or solving systems</p>
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<h3>4.Can vector equations describe curves?</h3>
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<h3>4.Can vector equations describe curves?</h3>
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<p>Yes. For example, the position of a basketball’s path during a game can be described using a vector<a>function</a>of time: r (t)=❰x(t), y(t), x(t)❱ </p>
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<p>Yes. For example, the position of a basketball’s path during a game can be described using a vector<a>function</a>of time: r (t)=❰x(t), y(t), x(t)❱ </p>
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<h3>5.How to find the intersection of two vector equations?</h3>
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<h3>5.How to find the intersection of two vector equations?</h3>
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<p> Make sure both sides of the vector are equal. Then solve for the parameters. If a solution exists, the lines intersect. </p>
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<p> Make sure both sides of the vector are equal. Then solve for the parameters. If a solution exists, the lines intersect. </p>
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<h2>Jaskaran Singh Saluja</h2>
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<h2>Jaskaran Singh Saluja</h2>
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<h3>About the Author</h3>
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<h3>About the Author</h3>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<h3>Fun Fact</h3>
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<h3>Fun Fact</h3>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>