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

1 Answer
Oct 21, 2015

Answer:

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#).