Hey, guys. So let's get started with our problem here. So we've got these 2 vectors a and b, and we're told some information about their scalar and vector products, their dot and cross products, and we want to calculate what their angle between them is. So in other words, what we're asked to find here is theta. But if you notice, I actually know nothing about these vectors. I don't know their magnitudes and directions or anything like that. So how do we figure this out? Well, if you're looking for theta, the one equation that pops up here with the theta term is going to be that the magnitude of the vector products, remember, this really just produces another vector. I'm just going to call it c. Is equal to absinθ, the two magnitudes and the angle between them. So, really, this is our target variable, and we actually know what this magnitude is. Right? Usually, we're asked to find it, but we actually know it's already 12. So if I try to rearrange this equation, what I'm going to end up with is that sinθ is equal to 12/ab. And I can't really do anything else with this. Right? So I can't even if I were to try to do this inverse sine, I still don't know what this a and b are. So because I don't know what the magnitude of those vectors, I can’t really finish off this equation. I'm going to need something else. I'm going to need another equation. Well, the one thing that we haven't seen or that we haven't used yet is that the scalar product is negative 8. Remember the scalar product is a dot b. So what I'm going to do here is I'm going to start out another equation, and I'm going to say that a·b is equal to remember that the definition of a dot b, it doesn't have a magnitude. It's just a number. And remember, it was just abcosθ, except we already know what the scalar product is. We know that a·b, whenever you do abcosθ, you're going to get negative 8. Alright? So here's what I'm going to do. I'm going to rewrite another equation. I'm going to get cosθ. Right? Because this is my other this is my target variable just in another equation, and this is going to be negative 8 divided by AB. So here's what I'm going to do. Right? I've got these two equations and they actually both have 3 unknowns. I've got the theta that's unknown and I've got my A and B that's unknown. So what I can do to get rid of them is I can actually divide the two equations. So what I'm going to do here is I'm going to do sinθ divided by cosθ. And what you're going to get here is that 12 divided by AB divided by negative 8 over AB. And what happens here is that the ABs are going to cancel. When you divide these two equations, the ABs cancel out and then basically what you end up with here is just a tangent theta. Right? Sine over cosine equals tangent. So you've got the tangent theta is equal to, and this is just 12 over negative 8, so this is just going to work out to negative 1.5. So we've got these two equations that had 3 unknowns and by dividing them, we were actually able to get down to only one equation with one unknown. So now what I have to do is just I just take the inverse tangent of this. And just make sure that your calculator is in degrees. So you're going to take the inverse tangent of negative 1.5. What you're going to get here is you're going to get negative 56.3 degrees. Alright. So is that our final answer? Well, remember that this angle here, this theta, is the angle between these two vectors A and B, but it has to be the smallest positive value. So what we can do here is if this angle here is negative 56, then we're going to have to add 180 degrees to it. One way you can kind of think about this, is that these two vectors have a scalar product of negative 8. So what happens here is I'm just going to draw like a sketch real quickly of what these vectors might look like. So this is a sketch. So this is my x and y axis. So if this vector is like let's say this is a, then in order for them to have a scalar product that's negative, it means that they have to be pointing in opposite directions. You only get scalar products that are negative whenever you have some components that point antiparallel. So, basically, what happens is we know that the angle between A and B has to be greater than 90 degrees. So this theta here must be greater than 90 degrees because a dot b is negative. It’s less than 0. Alright? So now that we've added a 180 to this, what you're going to get is 123.7 degrees, and that is the right answer. Alright? So now we know that this angle, this angle here, has to be 123.7. That's the only way you can get a vector product that has a magnitude of 12, but a scalar product that has a magnitude or or that's a scalar product that's negative 8. So hopefully, that makes sense. Let me know if you guys if you guys have any questions, and I'll see you in the next one.