Dot Product Calculator

Published on: March 8, 2026
Final Answer: Free • Full Steps: Pro

This Dot Product Calculator multiplies two vectors into a single number, and shows the working line by line. Matching components are multiplied and the results added, which measures how much of one vector lies along the other. A result of zero means the two are at right angles, and the sign tells you whether they broadly agree or oppose. Following the same pattern every time keeps each component paired with its partner.

Step-by-step method

  1. Set up the problem.
  2. Use the dot product formula A · B = a₁b₁ + a₂b₂ + a₃b₃.
  3. Substitute values, multiply, and add.
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. Use the dot product formula A · B = a₁b₁ + a₂b₂ + a₃b₃.
  3. Substitute values, multiply, and add.

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 - Use the dot product formula A · B = a₁b₁ + a₂b₂ + a₃b₃.

In this problem: Multiply matching components, then add the three products.

\(\vec{A} \cdot \vec{B} = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}\)

Step 3

Step 3 - Substitute values, multiply, and add.

In this problem: The dot product is \(32\).

\(\vec{A} \cdot \vec{B} = 14 + 25 + 36 = 4 + 10 + 18 = 32\)

Final Answer

\(\vec{A} \cdot \vec{B} = 32\)
See Example 2 Hide Example 2

Problem

\(\vec{A} = \left\langle \frac{1}{2},\; 0,\; 0 \right\rangle,\; \vec{B} = \left\langle 2,\; 3,\; 4 \right\rangle\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Use the dot product formula A · B = a₁b₁ + a₂b₂ + a₃b₃.
  3. Substitute values, multiply, and add.

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 \(\vec{B} = \left\langle 2,\; 3,\; 4 \right\rangle\).

\(\vec{A} = \left\langle \frac{1}{2},\; 0,\; 0 \right\rangle,\; \vec{B} = \left\langle 2,\; 3,\; 4 \right\rangle\)

Step 2

Step 2 - Use the dot product formula A · B = a₁b₁ + a₂b₂ + a₃b₃.

In this problem: Multiply matching components, then add the three products.

\(\vec{A} \cdot \vec{B} = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}\)

Step 3

Step 3 - Substitute values, multiply, and add.

In this problem: The dot product is \(1\).

\(\vec{A} \cdot \vec{B} = \frac{1}{2}2 + 03 + 04 = 1 + 0 + 0 = 1\)

Final Answer

\(\vec{A} \cdot \vec{B} = 1\)

Example Problems

Click one to load it into the box and solve it.