Dot Product Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Dot Product.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The dot product of two vectors \mathbf{a} = \langle a_1, a_2 \rangle and \mathbf{b} = \langle b_1, b_2 \rangle is the scalar \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2. Equivalently, \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta, where \theta is the angle between the vectors.

The dot product measures how much two vectors point in the same direction. If they point the same way, the dot product is large and positive. If perpendicular, it is zero. If they point in opposite directions, it is negative. Think of it as a 'similarity score' for directions.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The dot product converts two vectors into a single number that captures their directional alignment. Zero means perpendicular (orthogonal).

Common stuck point: The result of a dot product is a scalar (number), not a vector. This is the key difference from the cross product.

Sense of Study hint: Multiply matching components, then add all the products. If the result is zero, the vectors are perpendicular.

Worked Examples

Example 1

easy
Find \langle 1, 2 \rangle \cdot \langle 3, -1 \rangle.

Solution

  1. 1
    Step 1: Multiply corresponding components and sum: 1(3) + 2(-1).
  2. 2
    Step 2: = 3 - 2 = 1.
  3. 3
    Note: The result is a scalar (number), not a vector.

Answer

1
The dot product multiplies corresponding components and sums the results, producing a scalar. A positive dot product means the vectors point in roughly the same direction (angle < 90°).

Example 2

medium
Find the angle between \mathbf{a} = \langle 1, 0 \rangle and \mathbf{b} = \langle 1, 1 \rangle.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Are \langle 2, 3 \rangle and \langle -3, 2 \rangle perpendicular?

Example 2

hard
Find \langle 2, -1, 3 \rangle \cdot \langle 4, 5, -2 \rangle.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

vector operationsvector magnitude direction