If 𝐴 and 𝐵 are two perpendicular vectors, then 𝐴 dot 𝐵 is equal to what.
Another way of stating this problem is to ask what is the dot product of two perpendicular vectors. Recall that two vectors are perpendicular if they are at right angles to each other. We can sketch a pair of perpendicular vectors with the same initial point to help us.
As these vectors are perpendicular, they meet at a right angle, which we mark. The question is what is the dot product of these two vectors.
We have the geometric definition of the dot product which gives the dot product in terms of the magnitude of the two vectors in question and 𝜃, which is the measure of the angle between the two vectors.
We can’t assume anything about the magnitudes of the vectors. We’re not told anything in the question about them. But we can say something about 𝜃, the measure of the angle between the vectors.
The angle between them is a right angle. And so 𝜃 is 90 degrees. Substituting this value in, we find that when 𝐴 and 𝐵 are perpendicular, the dot product is the magnitude of 𝐴 times the magnitude of 𝐵 times the cosine of 90 degrees.
Hopefully, we recognize 90 degrees as a special angle and we remember the value of cos 90 degrees. It’s zero! And as we’re multiplying by cos 90 degrees on the right-hand side, the entire right-hand side is zero. And so the dot product of two perpendicular vectors is always zero.
This is one of the reasons, though not the only reason, why we care about the dot product. It gives us an easy way to see if two vectors are perpendicular.