00:01
The dot product of two vectors x and y is equal to the magnitude of each vector multiplied together times cosine of the angle theta, that is the angle between the two vectors.
00:16
So we want to find the angle theta, so we have x dot y divided by the product of their magnitudes is equal to cosine theta, so theta equals inverse cosine of x dot y divided by the product of their magnitudes.
00:44
So for an n -dimensional vector, this is equal to inverse cosine of x1 y1 plus x2 y2 plus x dot dot dot plus xn yn, okay, divided by the square root of x1 squared plus dot dot dot plus xn squared multiplied by the square root of y1 squared plus dot dot dot plus yn squared.
01:25
Okay, so we have our formula here.
01:27
Let's start with part a.
01:31
For a, we have x is negative 4 3, okay, that's two -dimensional, and y is 6 negative 1.
01:45
So then theta equals inverse cosine of negative 4 times 6 plus 3 times negative 1 divided by negative 4 squared plus 3 squared times the square root of 6 squared plus negative 1 squared.
02:15
I'm just going to simplify this and then i'll use a calculator.
02:21
So we have negative 24 minus 3, that's negative 27, divided by square root of 16 plus 9.
02:32
16 plus 9 is 25.
02:35
Square root of 25 is 5.
02:37
And then we have times the square root of 36 plus 1, which is 37.
02:44
Okay so now let's use a calculator to solve for theta.
02:49
Approximately 153 degrees.
02:54
Okay, this is the angle between vectors x and y.
02:56
Now let's do part b.
03:00
We have x is 1 negative 8 negative 3, and y is 0 negative 5 negative 2.
03:16
So let's use our formula.
03:19
Theta is inverse cosine of 1 times 0.
03:27
Okay, i'll do it like this.
03:28
1 times 0, so that's just 0.
03:33
Okay, plus negative 8 times negative 5, which is 40.
03:39
Plus negative 3 times negative 2, which is 6.
03:46
And then we divide by square root of 1 squared, which is 1, plus negative 8 squared, which is 64, plus negative 3 squared, which is 9, times square root of 0 squared, which is 0, plus negative 5 squared, which is 25, plus negative 2 squared, which is 4.
04:10
Okay, so this is inverse cosine of 46 divided by square root of 74 times square root of 29...