Hey, guys. So in earlier videos, I showed you how to calculate the vector product like a cross b by using this equation absinθ. So the magnitudes of the two vectors times the sine of theta, the angle between them. But in some problems, you'll need to calculate this vector product in terms of unit vector components, and you can't use this equation. For example, the example we're going to work out down here, we're going to write A cross B, but we're not given angles and directions. We're not even given a diagram. All we're given is these vectors in terms of their unit vectors, i's, j's, and k's. So we definitely can't use absinθ. So what I'm going to do in this video is I'm going to show you how to calculate the cross product by using components.

Now, right off the bat, I'm just going to mention that different textbooks and professors have their own method of doing this. If your professor has a really strong preference and wants you to learn it their way, do it that way. If not, I'm going to show you what I think is the easiest way to do this, and I'm going to show you a list of steps so you get the right answer every single time. So let's go ahead and jump right in.

Alright. So remember, the whole idea here is that this vector product A cross B just generates a new vector C. So what we need to do is we need to find a way to calculate its components. Remember, each vector, like whatever vector, can be written in terms of its x, y, and z components, I, j, and k. Now when we did this for scalar products and we did this, we didn't really have to calculate the components because all you had to do was just multiply the like components straight down and then add everything up together. But what you got was just a number. You didn't have to actually solve for any components. But C, this new vector here, actually does have components. So in other words, it can be written as cx⋅I+cy⋅j+cz⋅k. So we need to figure out a way to actually solve for cx, cy, and cz. So let's just jump right into our problem here because that's exactly what we're going to do. We're going to write the A cross B in terms of its components.

So let's just jump right into the first step here. The first thing you want to do is you're going to want to build a table of all of the X and Y and Z components for the two vectors that you have. But what you're going to do is you're going to repeat the X and Y columns twice. What I mean by this is you're going to build a little table that looks like this, and you have to extract the numbers for ax, ay, az and so on and so forth. If you look at the vector, this is i hat plus 2 j hats. In other words, this is a one. Right? It's kind of implied here, and this is a 2 in the y direction. So we have 12. Does A have a z component? Well, that would mean it has a k hat direction, but we have no k hat here. So we're just going to write a 0. There's no z component. And then you're just going to repeat this x and y twice. You're going to have 1 and 2. Now we're going to do the exact same thing for B. So this B has negative 2 in the x, and then it has 3 in the Y, and then it has 4 in the K. Then you just repeat the first two columns. So this is negative 2 and 3. So that's the first step. You just build out the table of components.

Now the next step here is we're going to write AB−BA for each component. Remember, we're really just trying to figure out what are cx, cy, and cz so we can write our vector in this form. So what happens here is I'm going to write AB−BA. I'm always going to leave a little space and you'll see why in just a second here. So AB−BA. Alright. So that's done.