Hey, guys. So hopefully you got a chance to check this one out on your own. We have these two vectors on our diagram here, and we want to find the magnitude and direction of the cross product. We're going to have this new vector , which is . 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 . Right? So the angle here is between and . The magnitude of is given as 12. The magnitude of is 8. Now we just need the sine of the angle between them. So, is it going to be this 30 degrees, or is it going to be something else? Well, this is where these three-dimensional diagrams can be tricky because they can mess up your perspective. So, this X-axis here is kind of back and forth. Right? So this XY plane is like the flat plane, and the Z-axis goes vertical. What happens is this vector might look like it's raised off the ground, but it's actually not. These dashed lines indicate that it's along the XY plane. So imagine that you have a tabletop like this, and imagine you have a vector pointing to the side except it's tilted away from you. So, that's what this vector is, and is pointing straight up. So, what is the angle between something that points horizontally and vertically? It's 90 degrees. It might not look like it, but this actually is, because this represents vertical and flat. The sine of 90 results in 1, and therefore the magnitude of our vector is going to be 96.

Now, how about the direction? For the direction, we're going to need to use the right-hand rule. The rule is for , you always point your fingers towards and curl towards . Take your right hand and point your fingers in the direction of . It looks like I'm moving towards you, but you're going to do it yourself. Your fingers are pointing away from you and then you curl your fingers up vertically towards . When you do that, your thumb will be pointing not straight at you but off to the right. This is our vector. It has a magnitude of 96 and it points here along this axis. This thumb, my vector, is still flat. It's still on the XY plane, just pointing in a different direction. We're going to use the same sort of dashed lines that I used for the other one to indicate that it's on this plane. We want to figure out what is the angle, or rather the positive angle, from the X-axis. So, what is this theta? The definition of this vector product is that it's perpendicular to both vectors that make it up. In other words, is perpendicular to the vertical, that's 90 degrees, and it's also perpendicular to the vector, that's also 90 degrees. If this is 30 degrees and this is 90 degrees, then the vector points off at 30 degrees as well. So, the answer is that here is equal to 30 degrees as a positive angle from the X-axis. Let me know if you have any questions.