WEBVTT
00:00:00.280 --> 00:00:05.640
If π΄ and π΅ are two perpendicular vectors, then π΄ dot π΅ is equal to what.
00:00:06.520 --> 00:00:11.400
Another way of stating this problem is to ask what is the dot product of two perpendicular vectors.
00:00:12.160 --> 00:00:15.720
Recall that two vectors are perpendicular if they are at right angles to each other.
00:00:16.240 --> 00:00:20.560
We can sketch a pair of perpendicular vectors with the same initial point to help us.
00:00:21.560 --> 00:00:25.520
As these vectors are perpendicular, they meet at a right angle, which we mark.
00:00:26.280 --> 00:00:29.680
The question is what is the dot product of these two vectors.
00:00:30.520 --> 00:00:40.520
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 π, which is the measure of the angle between the two vectors.
00:00:41.200 --> 00:00:43.640
We canβt assume anything about the magnitudes of the vectors.
00:00:44.040 --> 00:00:45.720
Weβre not told anything in the question about them.
00:00:46.240 --> 00:00:50.200
But we can say something about π, the measure of the angle between the vectors.
00:00:51.040 --> 00:00:52.600
The angle between them is a right angle.
00:00:53.040 --> 00:00:54.800
And so π is 90 degrees.
00:00:55.320 --> 00:01:04.920
Substituting this value in, we find that when π΄ and π΅ are perpendicular, the dot product is the magnitude of π΄ times the magnitude of π΅ times the cosine of 90 degrees.
00:01:05.960 --> 00:01:12.080
Hopefully, we recognize 90 degrees as a special angle and we remember the value of cos 90 degrees.
00:01:12.560 --> 00:01:13.080
Itβs zero!
00:01:13.680 --> 00:01:18.360
And as weβre multiplying by cos 90 degrees on the right-hand side, the entire right-hand side is zero.
00:01:19.160 --> 00:01:22.760
And so the dot product of two perpendicular vectors is always zero.
00:01:23.480 --> 00:01:27.840
This is one of the reasons, though not the only reason, why we care about the dot product.
00:01:28.280 --> 00:01:31.240
It gives us an easy way to see if two vectors are perpendicular.