What is the value of the dot product of two orthogonal vectors?

Oct 21, 2015

Zero

Explanation:

Two vectors are orthogonal (essentially synonymous with "perpendicular") if and only if their dot product is zero.

Given two vectors $\vec{v}$ and $\vec{w}$, the geometric formula for their dot product is

$\vec{v} \cdot \vec{w} = | | \vec{v} | | | | \vec{w} | | \cos \left(\theta\right)$, where $| | \vec{v} | |$ is the magnitude (length) of $\vec{v}$, $| | \vec{w} | |$ is the magnitude (length) of $\vec{w}$, and $\theta$ is the angle between them. If $\vec{v}$ and $\vec{w}$ are nonzero, this last formula equals zero if and only if $\theta = \frac{\pi}{2}$ radians (and we can always take $0 \le q \theta \le q \pi$ radians).

The equality of the geometric formula for a dot product with the arithmetic formula for a dot product follows from the Law of Cosines

(the arithmetic formula is $\left(a \hat{i} + b \hat{j}\right) \cdot \left(c \hat{i} + d \hat{j}\right) = a c + b d$).