00:01
All right, for this problem, we are given two vectors, v and w, and we're asked a couple questions about the plane that's spanned by these two vectors.
00:10
So we're asked to describe the plane, find a vector not in the plane, and then for these two given vectors, find out whether they are either in or not in the plane.
00:20
So we're going to answer these questions slightly out of order.
00:23
I would like to answer the second one first, to find a vector not in the plane, because that can be done relatively quickly.
00:30
This normal vector to the plane is the cross product of any two vectors in the plane.
00:41
So if we take this cross product, we are guaranteed to get a vector not in the plane.
00:48
So if you do this cross product, you should get that the normal vector is negative 3, negative 1, and 6.
01:00
All right.
01:02
Now, with this normal vector, we can also find an equation that describes the plane.
01:08
So to do that, we use the coefficients of the normal vector, or excuse me, we use the components of the normal vector as coefficients for the x, y, and z variables.
01:22
So here we have our negative 3x, negative 1y, and 6x, and this will equal some constant, which i will label as c.
01:30
And then we can use any point that's in the plane to figure out what this constant for c is.
01:36
So i'm going to use the w vector.
01:38
I'll let x equal 1, y equal 3, and z equal 1.
01:43
Plugging those in, we have negative 3 minus 3 plus 6 equals our constant c, which gives us that the constant is 0...