00:01
So for this question we have these points p, which is 3 minus 2 ,1, q which is 154, r which is 2 0 minus 6, and s which is minus 4, 15.
00:21
For part 8 it says, find the dot product of pq to rs.
00:29
So first thing we need to do is work out what pq and rs are, as we're just given these points.
00:36
So pq is going to effectively be the vector of p going to q.
00:41
So all we're going to do to make that is just going to be doing the values of q minus the values of p.
00:48
So that's going to be 1 minus 3, 5 minus minus 2, and 4 minus 1.
00:57
And for rs it's going to be going from r to s so we're going to do s the s is minus the r's so it's going to be minus 4 minus 2 1 minus 0 and 5 minus minus 6 so overall we have this are for both our pq and rs so we're going to have minus 2 7 3 dotted with minus 6 1 and 11.
01:32
So our dot products is just going to be multiplying the first value with this value, the second value with the second value, and third value with the third value.
01:43
So effectively, we're just multiplying all of these together.
01:45
So our dot product is just going to be equal to minus two times by minus six plus seven times by one plus three times by 11, which is 18 plus plus 33 which is equal to 58.
02:10
So that is our answer for part a and then for part b we have the angle between pq and rs.
02:19
Or for this we need to just use the the formula the dot product formula which is written as a dot b is equal to the magnitude of a times by the magnitude of b comes by cos theta.
02:38
Times by, cause theta, where theta is the angle that we want to find.
02:46
So therefore, theta is going to be equal to the arc cos, or if you are not familiar with arc cause, cause to the minus one, of, so it would be cos to the minus one on most calculators, the arc cause of a dot b over the magnitude of a times by the magnitude of b.
03:09
So, obviously in our case, our a is pq and r .b is rs.
03:16
So our angle theta, so we're going to say the angle pqrs, is equal to the arc cause of pq.
03:30
Pq dot rs, which are calculated in the last problem as 58, divided by the length of pq times by the length of rs.
03:41
Now to get the length of pq, so we have our vectors up here, to get the length of pq, all we need to do is just take the square root of all of the values in here squared.
03:54
So minus 2 squared, plus 7 squared, plus 3 squared.
03:59
So that's going to be equal to the square root of 4 plus 49 plus 9.
04:06
So that's going to be equal to the square root of 62.
04:16
So the square of 62 can be is as simple as we can go.
04:20
So we'll just leave it as square of 62.
04:22
Now for rs, our value for the magnitude of rs is going to be the exact same way.
04:31
The square of all of these values squared, so minus 6 squared plus 1 squared plus 11 squared.
04:37
That's going to be the square root of 36 plus 1 plus 1 2 1.
04:41
It's 11 squared is 1.
04:45
And so therefore our value for this is square root of 36 plus 1 plus 121, which is the square root of 158.
04:58
So on the bottom here we're going to have the arc cos of 58 over the square root of 62 times by the square root of 158.
05:11
So quite large numbers but it should be fine.
05:15
So the square root of 62 multiplied by the square root of 158.
05:21
And then we just take the arc cost of that.
05:23
So if we put that into a calculator, so cos the minus 1 of 58 over the square root of 62 times by the square root of 158, that gives us a value for the angle of theta as 54 .1 degrees.
05:39
And that is our answer for part b.
05:41
And then for part c, it says the scalar and the projection of ps along pr.
05:49
So if it's ps along pr, it's going to be a scalar of ps.
06:07
So ps will be up here onto pr.
06:12
If it says pr, yes.
06:22
Ps along pr.
06:24
Like so...