00:01
In this problem, we are going to use the concept of cross products of vectors in order to determine the area of a given triangle.
00:08
Now in this question, the three vertices pq and r of the triangle are given.
00:17
Now, first of all, what we need to do is find the vectors representing any two sides of the triangle.
00:23
So first, let us find the vector representing the side pq.
00:27
This is equal to the position vector of q which is 0, 3, 4, minus the position vector of p, which is 1 minus 1 2.
00:37
Now we obtain the difference of 2 vectors by subtracting the corresponding components.
00:42
So this is equal to 0 minus 1, 3 minus a minus 1, 4 minus 2.
00:48
So 0 minus 1 is minus 1, 3 minus 1 is 3 plus 1 which is 4 and 4 minus 2 is equal to 2.
00:55
Now i will assume this vector to be equals to you.
01:01
Using the same process we can find the vector representing the side pr that is equal to the position vector of r which is 618 minus the position vector of p which is 1 minus 1 2.
01:15
We will subtract the corresponding components so this is what we will obtain.
01:22
If we simplify this this is 6 minus 1 which is 5, 1 minus 1 which is 5.
01:27
Is 1 plus 1 which is 2 and 8 minus 2 which is 6.
01:33
I will assume this vector to be the vector v.
01:36
Now the area of the triangle formed by p, q and r will be equals to half of the magnitude, the modulus of the cross product u cross v.
01:47
So for that, first of all, we need to determine this cross product.
01:50
So first of all, we write a matrix.
01:52
The first row being the components of the vector u.
01:55
So that will be minus 1, 4 and 2.
01:57
The second row will have the components of the vector v, which is 5 to 6.
02:04
We use this in order to determine the cross product...