Subtraction of 2 Vectors
2026-02-28 06:03 Diff
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Last updated on August 5, 2025

The mathematical operation of finding the difference between two vectors is known as the subtraction of vectors. It helps in determining the relative position or direction of one vector with respect to another.

What is Subtraction of Vectors?

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

Vectors have two main components:

Magnitude: This is the length of the vector.

Direction: This indicates the orientation of the vector in space.

How to do Subtraction of Vectors?

When subtracting vectors, students should follow these steps:

Reverse direction: Change the direction of the second vector to find its additive inverse.

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

Resultant vector: The resultant vector represents the difference between the two original vectors.

Methods to do Subtraction of Vectors

The following are methods for subtracting vectors:

Method 1: Component Method

To apply the component method for vector subtraction, use the following steps.

Step 1: Break both vectors into their components.

Step 2: Reverse the direction of the second vector by changing the signs of its components.

Step 3: Add the components. Example: Subtract vector B from vector A.

Step 1: Write components of vectors A and B.

Step 2: Reverse the components of vector B.

Step 3: Add the components together. Answer:

Method 2: Graphical Method

When subtracting vectors graphically, draw the vectors with their tails at the same point. Reverse the direction of the second vector and then complete the parallelogram. The resultant vector is the diagonal.

For example, subtract vector B from vector A.

Solution: Draw vector A. Draw vector B with reversed direction. Complete the parallelogram to find the resultant vector.

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Properties of Subtraction of Vectors

In vector operations, subtraction has some characteristic properties:

  • Subtraction is not commutative In vector subtraction, reversing the order changes the result, i.e., A - B ≠ B - A.
  • Subtraction is not associative Involving three or more vectors, changing the grouping changes the result. (A − B) − C ≠ A − (B − C)
  • Subtraction as addition of the opposite Subtracting a vector is the same as adding its opposite, so vector subtraction can be viewed as vector addition with the direction reversed. A − B = A + (−B)
  • Subtracting zero vector leaves the vector unchanged Subtracting the zero vector from any vector results in the same vector: A - 0 = A.

Tips and Tricks for Subtraction of Vectors

Tips and tricks are useful for students to efficiently deal with vector subtraction. Some helpful tips are listed below:

Tip 1: Pay attention to both magnitude and direction when reversing a vector.

Tip 2: Use graph paper or vector tools to accurately draw vectors and find the resultant.

Tip 3: Break vectors into components to simplify calculations, especially in 2D or 3D space.

Forgetting to reverse direction

Students often forget to reverse the direction of the second vector when subtracting. Always remember to change the direction before performing vector addition.

Problem 1

Use the component method, (7, 2) - (3, 4) = (7 - 3, 2 - 4) = (4, -2)

Okay, lets begin

Subtract vector B (5, -3, 2) from vector A (1, 4, -1)

Well explained 👍

Problem 2

Use the component method of subtraction (1, 4, -1) - (5, -3, 2) = (1 - 5, 4 + 3, -1 - 2) = (-4, 7, -3)

Okay, lets begin

Subtract vector B (-2, 1) from vector A (3, -5)

Well explained 👍

Problem 3

(3, -5) − (-2, 1) = (3 + 2, -5 - 1) = (5, -6)

Okay, lets begin

Subtract vector B (6, 2, -4) from vector A (-1, 3, 5)

Well explained 👍

Problem 4

(-1, 3, 5) - (6, 2, -4) = (-1 - 6, 3 - 2, 5 + 4) = (-7, 1, 9)

Okay, lets begin

Subtract vector B (4, 0) from vector A (0, 4)

Well explained 👍

No, vectors must have the same dimensions to be subtracted.

1.Is subtraction commutative in vector operations?

No, the order of vectors matters in subtraction; changing them changes the outcome.

2.What are vector components?

Vector components are the projections of a vector along the coordinate axes, which help in simplifying vector addition or subtraction.

3.What is the first step of vector subtraction?

The first step is to reverse the direction of the second vector to find its additive inverse.

4.What methods are used for vector subtraction?

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

Common Mistakes and How to Avoid Them in Subtraction of Vectors

Subtraction in vector mathematics can be challenging, often leading to common mistakes. However, being aware of these errors can help students 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.