PROFESSOR: Find the volume of the parallelepiped formed by the vectors a equals 3i plus 4j minus k, b equals 2i minus j minus k, and c equals 3j plus k. To begin, we know that the volume is going to be equal to the absolute value of u dot parenthesis v cross w, close parenthesis, close the absolute value.

So if we have three vectors defining this u, v, and w, then that those works out. Now because we have a, b, and c, we can do similar things just, I'm just going to change this out. So the volume is going to be the absolute value of a dot parenthesis b cross c close parenthesis close the absolute value.

So let's first find the cross product of b and c. B cross c will be equal to, write this indeterminate notation i, j, k and then the vector b which is 2, negative 1, negative 1 and the vector c which is 0, 3, 1.

So we'd have this, the minor matrix of negative 1, negative 1, 3, 1, i minus the determinate of the matrix 2, negative 1, 0, 1 j rather plus the minor matrix 2, negative 1, 0, 3 k.

Now if we find the determinate of each of those matrices, this should come out to be 2i minus 2j plus 6k. Let me just double check my arithmetic here. OK. Yes. OK. That would be b cross c.

Now next, we need to find a dotted with b cross c. So just label this. Our first step is b cross c. Our second step is a dotted with b cross c, which is going to be the vector 3i minus 4j-- plus 4j, so 3i plus 4j minus k dotted with 2i minus 2j plus 6k.

And if writing in this notation makes you feel a little odd, it actually does me, I'm going to write this in component form. The angle bracket 3, 4, negative 1, angle bracket dot angle bracket 2, negative 2, 6.

Now multiplying some components here. So 3 times 2 plus 4 multiple negative 2 plus negative 1 multiplied by 6. And that should be 3 times 2 is 6. 4 times negative 2 is minus 8. And negative 1 times 6 is minus 6. So this should be minus 8. So our volume is the absolute value of a dot b cross c to the absolute value of negative 8. So that our volume is 8 units cubed.