Dot product

Definition

The dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. It is calculated as the sum of the products of corresponding entries.

5 Must Know Facts For Your Next Test

1. The dot product of vectors $\mathbf{a} = \langle a_1, a_2 \rangle$ and $\mathbf{b} = \langle b_1, b_2 \rangle$ is $a_1b_1 + a_2b_2$.
2. The dot product can be used to determine the angle between two vectors using the formula $\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}$.
3. If the dot product of two non-zero vectors is zero, the vectors are orthogonal (perpendicular).
4. The dot product is distributive over vector addition: $(\mathbf{a} + \mathbf{b}) \cdot \mathbf{c} = (\mathbf{a} \cdot \mathbf{c}) + (\mathbf{b} \cdot \mathbf{c})$.
5. In 3 dimensions, if $\mathbf{a}$ and $\mathbf{b}$ are given by $\langle a_1, a_2, a_3 \rangle$ and $\langle b_1, b_2, b_3 \rangle$, their dot product is $a_1b_1 + a_2b_2 + a_3b_3$.

Review Questions

• What does it mean if the dot product of two vectors is zero?
• How do you calculate the dot product for vectors in three-dimensional space?
• Which formula relates the dot product to the cosine of the angle between two vectors?

Related terms

Vector: A quantity defined by both magnitude and direction. Represented in algebra as an ordered pair or triple depending on dimensions.

Orthogonal Vectors: Two vectors are orthogonal if their dot product is zero. This means they are perpendicular to each other.

Magnitude: The length or size of a vector. Calculated as the square root of the sum of squares of its components.