Hi everyone. Welcome back to integralcalc.com. Today we're going to be doing another example
of cross product. This one we're given 2 vectors, a equals 2, -3 and b equals 4, 5.
And with an example like this, we're only given 2 direction numbers in each of our vectors,
we have to add to each of those a 0 to be the third coordinate to make this a full coordinate
in that space so we have like x, y and z. If it's not there, it's always going to be
0. We don't have to do any guess work about what that number is going to be or calculated.
It’s just a 0. So when we're only given two points like this, don't freak out. Just
add that 0. So that's the first thing that we're going to do.
And then the second thing that we're going to do in order to find the cross product,
we're multiplying a and b together to come up with one vector. The first thing we're
going to do in order to do that is build this matrix here. And when we build the matrix,
we'll always have i, a j and k in the first row. This never changes. We’ve got i, j
and k here in that order in the first row. And then from there, we go ahead and put the
direction numbers in vector a in our second row, 2, -3, 0 and the direction numbers in
vector b, 4, 5 and 0 in our third row. We always have a first and then b.
If we had a third vector, we can go ahead and put it under that. That’s how you build
our matrix. And then the second thing that we're going
to do is break that matrix apart into second order determinate matrices. It’s going to
be basically 2x2 matrices instead of 3x3. And the way that we do that, we're going to
need this formula up here. This matrix with the positive and negative signs and then we're
going to need to know one other thing. So the first thing that you’ll notice here,
we've got i, j and k. And we have a positive i, a negative j, and a positive k.
And the way that we know the signs, positive and negative on those three variables is based
on this formula up here. So when you've got this 3x3 matrix, you're going to want o build
yourself a 3x3 checkerboard pattern. You always start with a positive sign in the upper left-hand
corner and then build the rest of it with a checkerboard pattern so positive, negative,
positive and then start your second row with negative. Negative, positive, negative, positive
negative, positive. So you have this checkerboard pattern which matches this matrix down here
and they kind of like lay over the top of one other. And then you match them up and
use the sign on this matrix up here to determine what the sign is going to be on i, j and k.
And so as you can see, since i is on the upper left and we've got a positive sign on the
upper left, i here is going to be positive. Because j is in the second spot right here
and we have a negative sign there, j is going to be negative. And k is going to be positive
because k is in that last position and that's a positive sign based on our matrix that we
built. So that's how we know the signs of i, j and k.
And then to pull out this matrixes here, the way that we do it, this one here is multiplied
by i. So we're looking at i in this matrix here. So to build that 2x2 matrix we're going
to ignore everything in the same row and column as i. So you can see here. i is in the same
row as j and k, so we're going to ignore j and k completely. i is also in the same column
as 2 and 4 so we're going to ignore everything in this column, i, 2 and 4. And you can see,
if you block those off what we're left with is -3, 0, 5 and 0. And see we took those and
like transpose them into this 2x2 matrix. We’re left with -3, 0, 5 and 0.
So we take the four numbers that we're left with and we just move them just right over
into our 2x2 matrix. We don't switch on the numbers, we don't do anything. We just take
them, pick them up and move them over and drop them down.
They're multiplied by i here. We're going to do the same thing with j. So ignoring everything
in the same row and column as j means we're ignoring i, j, and k and it also means we're
ignoring j, -3 and 5. So we're left with 2, 0, 4 and 0. And even though 2 and 4 and 0
and 0 are separated, we just take them and move them straight over. So we still have
2, 0, 4 and 0. So we do the same thing for k, ignoring i,
j and k and k, 0 and 0. And we're left with 2, -3, 4 and 5. And that 2, -3, 4 and 5 come
over here and get multiplied by k. So that's how you break apart our matrix.
And the reason that we do that is so that we can get to this step here where we kind
of multiply everything out and we’ll be able to simply and solve for our cross product
vector. So the way that we do this, the way that we
get this equation here, we take the number in the upper left-hand corner and the number
on the lower right-hand corner and multiply them together. So negative 3 times 0 gives
us 0 and that number goes here, the number on our first spot and then we always subtract.
And then we take the number in the lower left-hand corner and the number on the upper right-hand
corner, diagonal like this and multiply them together. 5 times 0 is also 0 so we put 0
in that second spot. Let’s jump over to the example with k right
here so that we have some numbers that are different. We can make sure we've got it,
right? So remember upper left and lower right first, 2 times 5 is just 10 so 10 goes in
that first spot, always subtract and then 4 times 3 is -12 so we put that -12 right
here so we end up with a minus -12. So remember it's always going to be the upper
left times the lower right minus the lower left times the upper right. That goes for
i, j and k. So we built that, keeping the signs, the positive on the i, the negative
on the j, and the positive on the k, keeping those signs down here. And then it's just
a matter of simplifying. So the way that we simplify, I’ll go ahead
and simplify what's in the parenthesis here and then make it the coefficient on i, j and
k. So we end up with 0i minus 0j plus 22k because you've got a 10 minus a -12. It’s
just a positive 22 so we end up with a plus 22 k.
And you might think why would we keep the i and the j with the coefficients, they would
go away completely. And you could but it's easier to visualize if you leave the 0s because
what that tells us is that our final answer for our cross product is going to be 0, 0,
22. Because we pulled these three numbers from the coefficients on i, j and k that we
got when we broke apart that matrix and then multiplied everything out. So if you got rid
of i and j, it might be a little harder to see. If you just leave those coefficients,
you could just grab 1, 2 and 3, pull them right down into your final answer and our
cross product vector ends up being 0, 0, 22. And that's it. That’s our final answer.
So I hope that helped you, guys. And I’ll see you next time. Bye.