 Okay, in this video we're going to look again at the cross-product, but this time we're going to ask about its geometric meaning and Its properties when we come to manipulate it Okay, so if some vector c is the cross-product of two other vectors a and b We've already seen how to work that out, but what we can reasonably now ask is What does that vector c look like? You know, if we imagine a particular couple of vectors a and b there in space Where is this vector c? How is it related to them? We know how to work it out But what's its relationship with them? How should we think about it? And that's what we're gonna we're gonna figure out now. So we know that c is a vector So it has two properties. It has its magnitude and direction. Let's think about the magnitude first What is the magnitude of c and how does that relate to a and b? What is the length of that vector? It's pretty simple The magnitude of c is the magnitude of a times the magnitude of b times sign of the angle between a and b This is very similar to the dot product except with a sign instead of a cos So there we are. There's our two vectors a and b and an angle between them and From those Magnitudes the lengths of those two vectors and the angle we can work out the magnitude of c Note that if we cross a vector with itself the angle will be zero and so the cross product will be zero Just as we've already seen in our examples. That was easy enough. What about the direction of this new vector c? How does that relate? Okay, here's the thing the direction of c is perpendicular Sorry for my writing there wiggly writing c is perpendicular to both Vectors a and b so it's at right angles to each of those vectors separately and simultaneously What does that look like? Well, actually we can draw it in one of two ways one of which is right and one is wrong Let's just do that. So here's um, here's our vector a here's our vector b If we draw c like that and make it clear with this little symbol that it's at right angles to those two vectors That would be perpendicular to them both. How about this? We could also draw a vector a draw vector b again And we could go in the opposite direction simply literally the opposite direction and that would also be perpendicular to these two vectors one of these is actually strictly the correct case and the other is wrong by essentially a minus a Minus one multiple What's the way to work that out? So let's let's now figure that out There's actually a rule to remember it by it's called the right hand screw rule So let's draw that out kind of really clearly one more time We have two vectors a and b are going to say that a cross b is equal to some vector c That's fine So what we do is we put on the line Along which we know c must lie. So this is the line that's perpendicular to both a and b and we simply have to ask ourselves In in this picture does the vector c uh go upwards or does it go downwards? The trick is to write on the angle between a and b and give it a direction So that it's increasing from a to b it's the angle from a to b then you imagine taking your right hand and gripping that line in Such a way that your fingers curl in the same direction as the angle increases and then your thumb Points in the direction that the in the actual direction of c. Let's do another example Just to really make that clear. Here's a and b again So we know we need to be I've drawn these lying in a plane So we I'm now trying to draw a line that's perpendicular to that plane Vector c must lie in one direction or the other along this line. What do we do? We draw on the angle We now take our right hand and we imagine gripping that that line We've just drawn in such a way that our fingers curl In the direction in which the angle is increasing So it's like the anti clockwise direction in this picture And that's and then our thumb points in the correct direction for that vector. So it's in fact These are the two opposite cases Um, so that's the rule that allows you to construct the correct Uh direction for your vector geometrically geometrically Okay, uh, then let's just finally wrap up by thinking about the cross product and asking whether it has those properties that we looked at before For a vector addition the commutative property So for example is a cross b equal to b cross a It is not It is not equal to it Unlike the dot product unlike addition this one the cross product it matters the order and in fact It simply Introduces a minus sign If you swap the order of a and b so it's not commutative It nearly is in the sense that it gives you something similar it gives you The same thing up to a minus sign It's important to remember and you can just verify that by thinking about how we work out a and b with those diagonal products Now, how about the associative property? Can we say that a cross b cross c where b and c have already been worked out? It's the same as a cross b and then cross c Uh, what do we think is that going to work or not? In fact It uh, this is the associative property. You might ask whether this is true And the answer is no again the cross product does not have this property So the order in which you do your cross product if you have doing the cross product of three vectors does matter We can easily convince ourselves of this just by looking at a particularly Uh convenient example. Let's just use cartesian vectors i j k So let's just remind ourselves where these guys lie. They're perpendicular to each other i j and k Just our unit vectors going in the x y and z direction So suppose we have this guy i cross i cross k if we try evaluating it this way round With the i cross k being worked out first Well, that's just going to give us in fact minus j which you can confirm with the right hand rule that we just introduced And then that in turn will give us k. That's fine. So we've worked out In that instance, the answer is minus k now. Let's do it the other way round I cross i if we do that first That's just going to be zero because i cross i is zero. So it's game over already at that point So we can see two radically different answers here just depending on our order Finally, we could ask about the distributive property. So are we allowed to Multiply through using the cross product If we um if the second object in our cross product is a sum of two vectors. Can we do this? Well, uh, this at last is something that we are going to be allowed to do it is the distributive property And the cross product operation The vector product does have this property. We are allowed to do that But of course we must make sure to make to keep the order the same Okay, so I think that's everything for this video