Vector Subtraction Notes

Published on: September 29, 2026

This Vector Subtraction Calculator subtracts one vector from another in three dimensions, and shows the working line by line. Subtraction works component by component, because each axis is independent of the others. The result is the vector that points from the tip of the second to the tip of the first, which is why it measures the gap between them. Following the same pattern every time keeps the order of subtraction from being reversed.

Step-by-step method

  1. Set up the problem.
  2. Write the component-wise subtraction rule.
  3. Substitute the components into the rule.
  4. Subtract corresponding components to get the result.
See Example 1 Hide Example 1

Problem

\(\vec{A} = \left\langle 4,\; 5,\; 6 \right\rangle,\; \vec{B} = \left\langle 1,\; 2,\; 3 \right\rangle\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the component-wise subtraction rule.
  3. Substitute the components into the rule.
  4. Subtract corresponding components to get the result.

Step 1

Step 1 - Set up the problem.

In this problem: The components are \(\vec{A} = \left\langle 4,\; 5,\; 6 \right\rangle\) and \(\vec{B} = \left\langle 1,\; 2,\; 3 \right\rangle\).

\(\vec{A} = \left\langle 4,\; 5,\; 6 \right\rangle,\; \vec{B} = \left\langle 1,\; 2,\; 3 \right\rangle\)

Step 2

Step 2 - Write the component-wise subtraction rule.

In this problem: Vectors are subtracted component by component.

\(\vec{A} - \vec{B} = \left\langle a_{1} - b_{1},\; a_{2} - b_{2},\; a_{3} - b_{3} \right\rangle\)

Step 3

Step 3 - Substitute the components into the rule.

In this problem: Substitute each pair of components.

\(\vec{A} - \vec{B} = \left\langle 4 - 1,\; 5 - 2,\; 6 - 3 \right\rangle\)

Step 4

Step 4 - Subtract corresponding components to get the result.

In this problem: The resulting vector is \(\left\langle 3,\; 3,\; 3 \right\rangle\).

\(\vec{A} - \vec{B} = \left\langle 3,\; 3,\; 3 \right\rangle\)

Final Answer

\(\vec{A} - \vec{B} = \left\langle 3,\; 3,\; 3 \right\rangle\)
See Example 2 Hide Example 2

Problem

\(\vec{A} = \left\langle \frac{5}{2},\; 0,\; 0 \right\rangle,\; \vec{B} = \left\langle \frac{1}{2},\; 0,\; 0 \right\rangle\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the component-wise subtraction rule.
  3. Substitute the components into the rule.
  4. Subtract corresponding components to get the result.

Step 1

Step 1 - Set up the problem.

In this problem: The components are \(\vec{A} = \left\langle \frac{5}{2},\; 0,\; 0 \right\rangle\) and \(\vec{B} = \left\langle \frac{1}{2},\; 0,\; 0 \right\rangle\).

\(\vec{A} = \left\langle \frac{5}{2},\; 0,\; 0 \right\rangle,\; \vec{B} = \left\langle \frac{1}{2},\; 0,\; 0 \right\rangle\)

Step 2

Step 2 - Write the component-wise subtraction rule.

In this problem: Vectors are subtracted component by component.

\(\vec{A} - \vec{B} = \left\langle a_{1} - b_{1},\; a_{2} - b_{2},\; a_{3} - b_{3} \right\rangle\)

Step 3

Step 3 - Substitute the components into the rule.

In this problem: Substitute each pair of components.

\(\vec{A} - \vec{B} = \left\langle \frac{5}{2} - \frac{1}{2},\; 0 - 0,\; 0 - 0 \right\rangle\)

Step 4

Step 4 - Subtract corresponding components to get the result.

In this problem: The resulting vector is \(\left\langle 2,\; 0,\; 0 \right\rangle\).

\(\vec{A} - \vec{B} = \left\langle 2,\; 0,\; 0 \right\rangle\)

Final Answer

\(\vec{A} - \vec{B} = \left\langle 2,\; 0,\; 0 \right\rangle\)
See Example 3 Hide Example 3

Problem

\(\vec{A} = \left\langle 7,\; 3,\; -2 \right\rangle,\; \vec{B} = \left\langle 2,\; 5,\; 1 \right\rangle\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the component-wise subtraction rule.
  3. Substitute the components into the rule.
  4. Subtract corresponding components to get the result.

Step 1

Step 1 - Set up the problem.

In this problem: The components are \(\vec{A} = \left\langle 7,\; 3,\; -2 \right\rangle\) and \(\vec{B} = \left\langle 2,\; 5,\; 1 \right\rangle\).

\(\vec{A} = \left\langle 7,\; 3,\; -2 \right\rangle,\; \vec{B} = \left\langle 2,\; 5,\; 1 \right\rangle\)

Step 2

Step 2 - Write the component-wise subtraction rule.

In this problem: Vectors are subtracted component by component.

\(\vec{A} - \vec{B} = \left\langle a_{1} - b_{1},\; a_{2} - b_{2},\; a_{3} - b_{3} \right\rangle\)

Step 3

Step 3 - Substitute the components into the rule.

In this problem: Substitute each pair of components.

\(\vec{A} - \vec{B} = \left\langle 7 - 2,\; 3 - 5,\; -2 - 1 \right\rangle\)

Step 4

Step 4 - Subtract corresponding components to get the result.

In this problem: The resulting vector is \(\left\langle 5,\; -2,\; -3 \right\rangle\).

\(\vec{A} - \vec{B} = \left\langle 5,\; -2,\; -3 \right\rangle\)

Final Answer

\(\vec{A} - \vec{B} = \left\langle 5,\; -2,\; -3 \right\rangle\)
See Example 4 Hide Example 4

Problem

\(\vec{A} = \left\langle -1,\; 4,\; 6 \right\rangle,\; \vec{B} = \left\langle 3,\; -2,\; 2 \right\rangle\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the component-wise subtraction rule.
  3. Substitute the components into the rule.
  4. Subtract corresponding components to get the result.

Step 1

Step 1 - Set up the problem.

In this problem: The components are \(\vec{A} = \left\langle -1,\; 4,\; 6 \right\rangle\) and \(\vec{B} = \left\langle 3,\; -2,\; 2 \right\rangle\).

\(\vec{A} = \left\langle -1,\; 4,\; 6 \right\rangle,\; \vec{B} = \left\langle 3,\; -2,\; 2 \right\rangle\)

Step 2

Step 2 - Write the component-wise subtraction rule.

In this problem: Vectors are subtracted component by component.

\(\vec{A} - \vec{B} = \left\langle a_{1} - b_{1},\; a_{2} - b_{2},\; a_{3} - b_{3} \right\rangle\)

Step 3

Step 3 - Substitute the components into the rule.

In this problem: Substitute each pair of components.

\(\vec{A} - \vec{B} = \left\langle -1 - 3,\; 4 - \left(-2\right),\; 6 - 2 \right\rangle\)

Step 4

Step 4 - Subtract corresponding components to get the result.

In this problem: The resulting vector is \(\left\langle -4,\; 6,\; 4 \right\rangle\).

\(\vec{A} - \vec{B} = \left\langle -4,\; 6,\; 4 \right\rangle\)

Final Answer

\(\vec{A} - \vec{B} = \left\langle -4,\; 6,\; 4 \right\rangle\)