In the last segment, we saw how vectors can be multiplied to give a scalar value which is known as the dot product. In this segment, we are going to consider how vectors can be multiplied to give a vector with what is known as the cross product.
The cross product is denoted not surprisingly with a cross and so it's written as a cross b.
And the idea of the cross product is the area of the parallelogram spanned by a and b. So it is really given by the area of this parallelogram spanned by a and b and then instead of cosine phi, it involves the factor of sine phi. The magnitude is given by the magnitude of a, magnitude of b and instead of the cosine in the dot product it's sine of phi.
But the cross product also has a vector direction, which is given by the right hand rule. And again, the idea is the same. You take your right hand, you curl it from the vector a to the vector b. And you're curling your hand from the vector a to the vector b.
And it's where your thumb points. That is where the direction of the cross product is. An interesting point is the way a and b, in which order they come matters unlike for the dot product. So if I were to do b cross a, I would have to take my hand, point it at b, curl it around to the vector a and now my thumb would be pointing down.
So it would be pointing in the opposite direction as a cross b. Therefore we have this negative formula, they are not commutative. a cross b is equal to minus b cross a.
But the very interesting point is that the cross product is orthogonal to both of the initial vectors and therefore it is very useful in creating a coordinate system which we will talk about in a later segment.
You can visualize the X, Y and Z axes. And by using the right hand rule, you can yourself verify that these formulae are true. That X cross Y is equal to Z. Y cross Z is equal to X. And Z cross X is equal to Y.
It's easy to memorize this by just thinking about the sequence x, y and z and cyclically if you go around it, x cross y will be z, y cross z will be x, and z cross x will be y. If you flip the order, you negate the sign.
Here are some properties of the cross product which you can consider here. a cross b is equal to minus b cross a.
Cross product of a vector with itself because the sign is 0 is 0, we have associativity, a cross (b + c) is equal to (a cross b) + (a cross C) and scaling a vector by scalar, scales the whole cross product. Those are the basic properties of the cross product.
The formula for the cross product can be derived. When we derive the formula, what we want to do is consider a vector that's given by x_a times X, plus y_a times Y, plus z_a times Z.
And now we want to take the cross product of this vector, so I'll just write this as a cross product with the same thing. So we had x_b, y_b and z_b, and so we can consider the vectors, X, Y, and Z. Okay, so now if you take the cross product because of associativity you can consider all of the diffrent nine terms in the cross product. Let's consider some possibilities, so if you consider X cross X, but you know that X cross X is equal to zero, we know that X cross product with X is going to be equal to 0, as is Y cross Y and Z cross Z. That's because any vector cross product with itself, the sign is equal to 0. So the terms x_a * x_b, y_a * y_b and z_a * z_b will disappear.
Now we can consider the X cross Y term so X cross Y here and I can write that as x_a times y_b. It involves X cross Y, but we know that X cross Y is equal to Z.
And so I write this as the vector Z. Equivalently here, we have Y cross X, so y_a * x_b. But now we have Y cross X. And we know that Y cross X is equal to -X cross Y. So we can put this as a negative sign, and this is the vector Z. So now we have (x_a * y_b - y_a * x_b) * Z.
So, I can write this down. There are similar kinds of results for X and Y.
Which allows us to create the formula shown here. So a cross b is written in this way in terms of a determinant. So in the determinant the first value X involves (y_a * z_b - z_a * y_b). So (y_a * z_b - y_b * z_a).
The next value Y involves (z_a * x_b - x_a * z_b). And Z involves (x_a * y_b - y_a * x_b). So (x_a * y_b - y_a * x_b). Normally determinants are used to evaluate numbers but in this case it's a simple way of doing it where you note that the X, Y and Z here are vectors.
And similarly here this corresponds to X, this corresponds to Y and the last row corresponds to Z.
What is even more interesting is that we can write this in terms of a matrix-vector multiplication where a cross b is some matrix times the column vector for the vector b. So the vector b is x_b, y_b and z_b.
And we are writing it with this matrix here, you can verify by matrix vector multiplication that it does in fact work out. This matrix which has this interesting form of zeroes along the diagonals and anti symmetry. So za za, but change in signs is what is known as the dual matrix of the vector a. So given a vector a we can convert it into this dual matrix, "A star" (A*), which can be used then to compute the cross product with the vector b.