Subtraction of Vectors
2026-02-28 10:46 Diff
https://brightchamps.com/en-us/math/numbers/subtraction-of-vectors

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Last updated on September 8, 2025

The mathematical operation of finding the difference between two vectors is known as the subtraction of vectors. It helps in understanding relative positions, directions, and magnitudes in physics and engineering problems.

What is Subtraction of Vectors?

Subtracting vectors involves adding the additive inverse of the second vector to the first. This involves reversing the direction of the second vector and then performing vector addition.

A vector has three components:

Magnitude: The length or size of the vector.

Direction: The orientation of the vector in space.

Components: The projections of the vector along the coordinate axes.

How to do Subtraction of Vectors?

When subtracting vectors, follow these rules:

Reverse direction: Reverse the direction of the vector being subtracted by changing its sign.

Add vectors: Perform vector addition by adding the corresponding components of the vectors.

Simplifying result: The result is a new vector representing the difference between the original vectors.

Methods to do Subtraction of Vectors

The following are methods of subtracting vectors:

Method 1: Graphical Method

To apply the graphical method for vector subtraction, use these steps:

Step 1: Draw the first vector using appropriate scale and direction.

Step 2: Draw the second vector from the head of the first vector but in the opposite direction.

Step 3: The resultant vector from the tail of the first vector to the head of the second vector is the difference.

Example: Subtract vector B from vector A.

Step 1: Draw vector A.

Step 2: Draw vector B starting from the head of A but in the opposite direction.

Step 3: The resultant vector from the start of A to the end of the reversed B is A - B.

Method 2: Analytical Method

Subtract vectors using their components. Write the vectors in component form and subtract the corresponding components.

Example: Subtract B = <2, -3> from A = <5, 4>.

Solution: A - B = <5 - 2, 4 - (-3)> = <3, 7>

Therefore, the resultant vector is <3, 7>.

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Properties of Subtraction of Vectors

Vector subtraction has certain properties:

  1. Subtraction is not commutative Changing the order of vectors changes the result, i.e., A - B ≠ B - A.
  2. Subtraction is not associative When three or more vectors are involved, changing the grouping changes the result. (A − B) − C ≠ A − (B − C)
  3. Subtraction as addition of the negative Subtracting a vector is the same as adding its negative, converting subtraction into addition by reversing the vector's direction. A − B = A + (−B)

Tips and Tricks for Subtraction of Vectors

The following tips are helpful for vector subtraction:

Tip 1: Always pay attention to the direction before subtracting vectors.

Tip 2: For ease, convert vectors into component form and handle subtraction as component-wise operations.

Tip 3: Use vector diagrams for visual learners to better understand direction changes and resultant vectors.

Forgetting to reverse direction

Students often forget to reverse the direction of the vector being subtracted. Always remember to reverse the direction of the vector before adding.

Problem 1

Use the component method: A - B = <6 - 2, 3 - 1> = <4, 2>

Okay, lets begin

Subtract vector D = <4, -5> from vector C = <7, 2>

Well explained 👍

Problem 2

Use the component method: C - D = <7 - 4, 2 - (-5)> = <3, 7>

Okay, lets begin

Subtract vector F = <-3, 2> from vector E = <1, -4>

Well explained 👍

Problem 3

E - F = <1 - (-3), -4 - 2> = <4, -6>

Okay, lets begin

Subtract vector H = <0, 3> from vector G = <5, 5>

Well explained 👍

Problem 4

G - H = <5 - 0, 5 - 3> = <5, 2>

Okay, lets begin

Subtract vector J = <2, -3, 1> from vector I = <4, 1, -2>

Well explained 👍

No, vectors must have the same dimensions to be subtracted; they must have the same number of components.

1.Is vector subtraction commutative?

No, the order of vectors affects the outcome; reversing the order changes the result.

2.What are the vector components?

Vector components are the projections of a vector along the coordinate axes, such as x, y, and z components in 3D space.

3.What is the first step in vector subtraction?

The first step is to reverse the direction of the vector being subtracted by negating its components.

4.What methods are used for vector subtraction?

The graphical method and the analytical (component) method are used for subtracting vectors.

Common Mistakes and How to Avoid Them in Subtraction of Vectors

Subtracting vectors can be tricky, leading to common mistakes. Awareness of these errors can help avoid them.

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.