2.10 Vector Subtraction

Definition

Vector subtraction is the process of finding the difference between two vectors.

Instead of subtracting directly, vector subtraction is performed by adding the opposite of a vector.

The vector $-\vec{B}$  has:

• the same magnitude as $\vec{B}$  
• the opposite direction

How to Subtract Vectors

Follow these steps:

1. Draw vector $\vec{A}$
2. Reverse the direction of vector $\vec{B}$ to obtain $-\vec{B}$
3. Add $-\vec{B}$ to $\vec{A}$ using the head-to-tail method.
4. Draw the vector from the tail of $\vec{A}$ to the head of $-\vec{B}$

Subtracting Vectors Using Components

Vectors can also be subtracted by subtracting their corresponding components.

If

and

then

Therefore, the x-components are subtracted:

and the y-components are subtracted:

Components along different axes cannot be subtracted directly. An x-component must be subtracted from another x-component, and a y-component must be subtracted from another y-component.

Example

Two displacement vectors are given by:

Subtracting the corresponding components:

The difference vector has an x-component of 1 m in the negative x-direction and a y-component of 3 m.

Key Points

• Vector subtraction is performed by adding the opposite vector.
• The opposite vector has the same magnitude but the opposite direction.
• Reverse the direction of the vector before performing head-to-tail addition.