Vector Algebra
2026-02-28 01:02 Diff
https://brightchamps.com/en-us/math/algebra/vector-algebra

Zero Vector

Definition: The zero vector has a magnitude of zero with no specific direction.
Notation: 0
Example: \(0 = 0i  + 0j + 0k\)

Unit Vector 

Definition: We use this to indicate direction, and a unit vector has a magnitude of 1.
Notation: A
Example: For \(A = 3i +4j, A = 32 + 42 = 25 = 5\)

Position Vector 

Definition: A position vector shows the location of a point in space relative to the origin. It shows the arrow pointing from the origin to the point.
Notation: r
Example: The position vector of a point P(x, y, z) is \(r = xi  + yj + zk \)

Co-initial Vectors

Definition: They have the same initial point, even if they point in different directions or have different magnitudes. 
Example: If Vector A and Vector B both start from the origin (0, 0) but point in different directions, then they’re co-initial vectors. 

Collinear Vector

Definition: These lie near the same line or are parallel to each other.
Example: \(A = 2i + 3j\) and \(B = 4i + 6j\) are collinear.

Equal Vector 

Definition: These have the same magnitude and directions at any point of their initial points.
Example: \(A = 3i  + 4j and B = 3i  + 4j\)

Negative of a Vector

Definition: This has the same magnitude as a given vector but points in  the opposite direction.
Example: If \(A = 5i  + 2j\), then the negative of vector A is:
\(-A = -5i  - 2j\)

Parallel Vectors 

Definition: Parallel vectors have the same or exactly opposite direction. They may have different magnitudes, but their directions are aligned. 
Example: Vector A and KA, where k is a scalar, are parallel.

Orthogonal Vectors

  Definition: These vectors are perpendicular to each other, meaning they form a 90o degree angle when they intersect. 
Example: If \(A = i  + 2j\) and \(B = -2i  + j\), then \(AB = 0\).

 Coplanar Vector

Definition: This lies in the same plane.
Example: Vectors \(A = 2i  + 3j\) and \(B = -i  + 4j\) are coplanar with any vector in the xy-plane.