3. Vectors

Intro to Cross Product (Vector Product)

# Find the Vector Product

Patrick Ford

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Hey guys, so hopefully got a chance to check this one out on your own. So we have these two vectors on our diagram here and we want to find the magnitude and direction of the cross product. So we're gonna have this new vector C. Which is a cross B. So let's go ahead and get started. Now I can't draw this vector on this diagram because I don't know what the direction is, but I can go ahead and find out the magnitude by using my equation for the cross product. The magnitude is just a. B. Times the sine of the angle between them, right? So stated here is between A and B. The magnitude of A is just giving us 12, the magnitude of B. Is eight. Now we just need the sine of the angle between them. So it's gonna be this 30 degrees, It is gonna be something else. Well this is where sort of these three D diagrams can be kind of tricky because because you sort of can mess you up on the perspective of things. So this X axis here is kind of like back and forth. Right? So this xy plane is kind of like the flat plane and the Z plane or the z axis goes vertical. So what happens is this a vector? It might look like it's kind of raised off the ground, like it's kind of like doing this but it's actually not. So what happens here is these dash lines are supposed to indicate that it's along the X. Y plane. So imagine that you have sort of a tabletop like this and imagine you have this vector that's pointing to the side except it's kind of tilt away from you so it's kind of like tilting away like that so that's what this A vector is, and then B. Is just pointing straight up. So what is the angle between something that points horizontal and vertical? It's 90 degrees. So it might not look like it's 90 degrees. But this actually is because sort of this is like vertical and flat. Alright, so this is actually 90 degrees here, so that's the sign of 90. And so therefore the magnitude of our vector is going to be 96. Alright, so that's the magnitude have about the direction. Well for the direction we're gonna need to use the right hand rule. Remember the rule is for a cross B. You always point your fingers towards A. And you curl towards B. So here's what I want you to do, don't you take your right hand and point your fingers in the direction of A. So what happens is it's kind of flat like this. But remember you're gonna have to turn your fingers away from you, it looks like I'm moving towards you but I want you to do it yourself, your finger's gonna be pointing away from you and then you're gonna curl your fingers sort of up vertically towards B. So when you do that, what happens is your thumb is going to be pointing not straight at you, but kind of off to the right like this. So it's gonna be kind of looking like this all right now, what we want to do is this is our c vector. It's a it has a magnitude 96 it points here along this axis. Now this thumb, my c vector is still flats, right? It's still flat, it's just still on the xy plane, just pointing in a different direction. So I'm gonna use the same sort of dash lines that I used for the other one to kind of indicate that it's on this plane. What I want to do is essentially figure out what's the angle of the positive angle um from the X axis, remember? So that's this one over here. So basically want to figure out what is this data now? So, which kind of equation can we use remember this data? Is is the angle between A and B. Tells us nothing about this angle over here. So we can't use that equation. Well, it turns out we're gonna have to use the property of the of the vector product. Remember that? The definition of this vector is that it's perpendicular to both of the vectors that make it up, in other words, C is perpendicular to the vertical, that's 90 degrees and it's also perpendicular to the A vector that's also 90 degrees. So what happens is I'm gonna look at these two vectors over here because I'm told some information about this 30 degrees. And this angle. So what I'm gonna do is I'm gonna try to try to draw sort of like a top down view of this diagram and this is what I end up with. Right? So you end up with a little diagram looks like this. This is your plus Y and your plus X. Right? So basically I'm just imagining that I'm kind of just looking down at the table top from above. So I've got my vector a. That points over here, this is a equals 12. We know this is 30 degrees and this is my C vector, this is my new cross products, right? So it looks like this actually when we look it's gonna be like that. Right? So this see here is 96. Alright, So this angle here, Theta is actually just this angle. The angle between this X axis over here and see. All right. So how do we find that? Well, we know that The difference between X&Y. This angle here is 90. And if we know that this here is 30° and we know that this is also a 90° angle because the vector product is perpendicular. Then what happens here is if this is 90, then this angle also has to be 30°. So we have 30 and then we have 90 over here. And that means that this vector points off at 30 as well. So, that's actually your answer. So the answer is that data here is equal to 30 degrees as a positive angle from the X axis. Alright, so hopefully that makes sense guys, let me know if you have any questions.

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