‫So all that was for the vector's in three dimensions, which is exactly what we need for discourse.
‫However, you can also compute the cross product mathematically.
‫Let's see how to do it.
‫Suppose that you have a vector V1 and vector V two, then you would compute the one vector, cross the
‫two vector, or you can write it in this form.
‫Then you would compute the cross product like this.
‫You create a three by three matrix where the first row will be this one and these are the unit vectors.
‫Then the second row will be the one vector.
‫But if here it's as a column vector here you write it as a row vector and the third row will be the
‫V two vector.
‫But now as our role vector and by the way, the unit vectors I j and K, their purpose is to give you
‫a direction.
‫They're magnitudes are all one.
‫So I has a magnitude of one, J has a magnitude of one and K has a magnitude of one.
‫But what they do they tell you in which direction you're going.
‫So for example, this V one vector, I can also write it down like this.
‫So this vector here that I have written in the form of column vector, I can write it like this v one
‫vector equals V one X I plus V one Y J and plus V one Z, K.
‫This formulation here is equivalent to this formulation here.
‫And so this I J and K, their unit vectors, they are all equal to one.
‫But you know that if you see I then you know that this component is in the X direction and thanks to
‫J you know that this component is in the Y direction.
‫And thanks to K you know that this component is in the Z direction.
‫And we're going to talk about the unit vectors more later in this section as well.
‫And so let's go back to computing our cross product.
‫We have this three by three matrix that we have constructed like this.
‫And now what we do, we take the determinant of it, which looks like this here.
‫I list the I J and K unit vectors horizontally and I assign the X components of the V one and the two
‫vectors to I like this then the Y components of these two vectors I assigned to J and the Z components
‫of them I assign to K and then I compute the determinant of the matrix.
‫The formula for the determinant comes from linear algebra and it looks like this.
‫You take the eye vector and you cross the corresponding row and column and the two by two matrix that
‫is remaining, you take its determinant and that will be your eye or X component in your total cross
‫product.
‫So it will be like that.
‫You take the eye vector and then to compute the determinant of a two by two matrix, you first multiply
‫the one Y times V to Z and then you subtract V, one Z times V to Y.
‫Let's just call it a four now and then you do the same thing for J and you cross out the corresponding
‫row and column your remaining two by two matrix will be this one composed of these elements here.
‫But in case of J there is an exception.
‫You have to put minus in front of J.
‫So the J component of your total cross product will be this one, this entire thing.
‫And let's call this part B, and now the only thing that's left is to find the key part of this total
‫cross product, and that will be your exercise.
‫Why don't you find the key part for this cross product and you'll see the solution in the next video?