Question

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

Answer 1

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) * vec(w)=||vec(v)|| ||vec(w)|| cos(theta)`, 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=pi/2` radians (and we can always take `0 leq theta leq 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 `(a hat(i)+b hat(j)) * (c hat(i) + d hat(j))=ac+bd`).