Vector Addition Notes

Published on: September 29, 2026

This Vector Addition Calculator adds two vectors in three dimensions, and shows the working line by line. Vectors add component by component, because movement along x is independent of movement along y or z, so the three directions never mix. The result is a single vector that reaches the same place as the two taken one after the other. Following the same pattern every time keeps each component paired with its own axis.

Step-by-step method

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

Problem

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

Approach

Step-by-step method

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

Step 1

Step 1 - Set up the problem.

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

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

Step 2

Step 2 - Write the component-wise addition rule.

In this problem: Vectors are added 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.

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

Step 4

Step 4 - Add corresponding components to get the result.

In this problem: The resulting vector is \(\left\langle 5,\; 7,\; 9 \right\rangle\).

\(\vec{A} + \vec{B} = \left\langle 5,\; 7,\; 9 \right\rangle\)

Final Answer

\(\vec{A} + \vec{B} = \left\langle 5,\; 7,\; 9 \right\rangle\)
See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

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

Step 1

Step 1 - Set up the problem.

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

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

Step 2

Step 2 - Write the component-wise addition rule.

In this problem: Vectors are added 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.

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

Step 4

Step 4 - Add corresponding components to get the result.

In this problem: The resulting vector is \(\left\langle 3,\; 0,\; 0 \right\rangle\).

\(\vec{A} + \vec{B} = \left\langle 3,\; 0,\; 0 \right\rangle\)

Final Answer

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

Problem

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

Approach

Step-by-step method

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

Step 1

Step 1 - Set up the problem.

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

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

Step 2

Step 2 - Write the component-wise addition rule.

In this problem: Vectors are added 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.

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

Step 4

Step 4 - Add corresponding components to get the result.

In this problem: The resulting vector is \(\left\langle 5,\; 4,\; 2 \right\rangle\).

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

Final Answer

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

Problem

\(\vec{A} = \left\langle -3,\; 0,\; 7 \right\rangle,\; \vec{B} = \left\langle 6,\; 2,\; 1 \right\rangle\)

Approach

Step-by-step method

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

Step 1

Step 1 - Set up the problem.

In this problem: The components are \(\vec{A} = \left\langle -3,\; 0,\; 7 \right\rangle\) and \(\vec{B} = \left\langle 6,\; 2,\; 1 \right\rangle\).

\(\vec{A} = \left\langle -3,\; 0,\; 7 \right\rangle,\; \vec{B} = \left\langle 6,\; 2,\; 1 \right\rangle\)

Step 2

Step 2 - Write the component-wise addition rule.

In this problem: Vectors are added 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.

\(\vec{A} + \vec{B} = \left\langle -3 + 6,\; 0 + 2,\; 7 + 1 \right\rangle\)

Step 4

Step 4 - Add corresponding components to get the result.

In this problem: The resulting vector is \(\left\langle 3,\; 2,\; 8 \right\rangle\).

\(\vec{A} + \vec{B} = \left\langle 3,\; 2,\; 8 \right\rangle\)

Final Answer

\(\vec{A} + \vec{B} = \left\langle 3,\; 2,\; 8 \right\rangle\)