Dot Product - Yousef's Notes

The dot product of two vectors $\mathbf{a} = \begin{pmatrix} a_1 \ a_2 \ a_3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} b_1 \ b_2 \ b_3 \end{pmatrix}$ is defined as: $$ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 $$

The dot product is a scalar value that represents the amount of “similarity” between the two vectors. It is used to:

Calculate the magnitude of a vector: $\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2$
Determine the angle between two vectors: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta)$
Project one vector onto another: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta)$