Cross Product: Study with Video Lessons, Practice Problems & Examples

The cross product of two vectors results in a new vector that is perpendicular to the original vectors. To compute the cross product, set up a matrix with the unit vectors i, j, k in the first row, followed by the components of the vectors. Use the determinant method to find each component by multiplying and subtracting the appropriate elements. The resulting vector can be expressed in component form. Recognizing the pattern of multiplying unlike components simplifies the process, making it easier to derive the final vector result.

Computing the Cross Product

Welcome back, everyone. So you may recall in a previous video, we talked about how to find the dot product between 2 vectors. And this was an operation where you took 2 vectors and multiplied them together, and it gave you a scalar result. Well, in this video, we're going to be learning about something called the cross product. And just like the dot product, the cross product is a way to multiply 2 vectors together, except rather than getting a scalar result, you actually end up getting a vector result. Now this process for finding the cross product is pretty tedious, but we're going to be learning in this video that there actually are some patterns you can recognize in this operation. So let's just jump right into things. Now, when doing a cross product, you're taking 2 vectors and you're multiplying them. And the other vector that you get is always going to be perpendicular to the original vectors. So if we have these vectors, u and v, and we want to find their cross product, the cross product of u and v is going to be a vector that points perpendicular to the vectors that we already have. So it would look something like that. Now, like I mentioned, doing this cross product can be a pretty tedious process. So to make sure we understand this well, we're going to solve this by the steps:

Let's say we have this example where we're told to find the cross product w, which is equal to the cross product of u and v. Now I can see that we have vector u = 2 0 1 and vector v = 0 -1 2. Now our first step with the cross product should be to write a matrix for the i, j, k components for each vector. Now in this matrix, the i's, j's, and k's go on top. Now in the second row here, we're going to have our vector u. And notice how vector u is the first thing that we see in this cross product operation. So we're going to take the first vector we see, we're going to write its components. So we have the i component which is 2, we have the j component which is 0, and then we have the k component which is 1. So that's going to be vector u. Now below that we're going to write the components for vector v, which I can see here are 0, negative 1, 2. So this is our first step. Now our second step is going to be to repeat i and j columns outside of the matrix. So I'm going to repeat the i column, which I can see is 2 0. I'm also going to repeat the j column, which I can see is 0 negative 1. So that's our second step. Now, our third step is going to be for each of the w components, which are these components right here, to write uv-vu for each of them. So this is going to be the pattern for each component. We're always going to have uv-vu, and that's the same for the z component. It's going to be uv-vu. Now the way we do the cross product operation is by using this matrix and multiplying down then up for each product. And that's going to create this kind of cross pattern, which you can think of it like a cross product. So we're going to start with I. This corresponds to the x component of our vertex. If we start with I, we can cross down through here, crossing through the j component, and then we can cross up right there. This is what the pattern is going to look like. Now notice when I do this, I have the y component multiplied by the z component, and then I also have the y component here multiplied by the z component there. So ends up being yzyz. Now let's move on to the j unit vector which corresponds with the y component. Now if I start with j, I can use the same cross pattern where I first cross down, and then I cross up. Now notice when I do this with the j, I end up with the z component times the x component, and then I once again end up with the z component times the x component. That is going to be the y component. Now what I'm going to do from here is move on to the k component. And for k I'll just use the same crossing pattern. I'll cross down first and then I'll cross up. So we're going to have x times y and then we're going to have x times y. Now notice the pattern that happens here. Notice each time we did this cross down and up strategy, we always ended up with the unlike components that got multiplied together. So notice for the x component, we had yzyz. Notice for the y component, we had zxzx, and then notice for the z component, we had xyxy. So as you can see, when you do this cross product operation, you start to recognize a pattern. Now, we can go ahead and actually figure out what our solution is now that we know what these components are. So we're going to have 0 times 2, which is 0, minus negative 1 times 1, which is negative 1. And 0 minus negative 1 is the same thing as 0 plus 1, which comes out to just 1. That's the x component. Now for the y component, well, let's go to j. For j, we're going to have 1 times 0, which is 0, minus 2 times 2, which is 4. 0 minus 4 is negative 4, and that's our y component. Now let's move on to the k component. For k, I can see that we have 2 times negative 1, which is going to be negative 2 minus 0 times 0, which is 0. Negative 2 minus 0 is negative 2. So now we have our x, y, and z components, so I just need to write them in this vector. So I can see that we're going to have 1 I, then negative 4 j, and then negative 2 k. So this right here is what our vector is going to look like, and if I want to write this in the component form, it's going to be 1, negative 4, negative 2. So this right here is the solution to the problem, and that is how you can find a cross product. So as you can see, it's pretty tedious, but by doing this cross down and up strategy on this matrix, notice we were able to find it was the unlike components being multiplied each time. So if you just subtract those results like we did here, it will allow you to find the vector that you're looking for. So I hope you found this video helpful. Thanks for watching and please let me know if you have any questions.

If vectors $v ⃗=⟨3,1,0⟩$, $u ⃗=⟨0,-2,0⟩$, and $w ⃗=v ⃗×u ⃗$, find $w⃗⃗$.

• A. $w⃗=\langle0,0,-6\rangle$
• B. $w⃗=\langle0,-2,0\rangle$
• C. $w⃗=\langle0,0,6\rangle$
• D. $w⃗=\langle0,0,-2\rangle$

If vectors $a⃗=5î$, $b⃗=12k̂$ and $c⃗=a⃗\times b⃗$ , find $c⃗$.

• A. $c⃗=60ĵ$
• B. $c⃗=-60ĵ$
• C. $c⃗=5î+12k̂$
• D. $c⃗=60k̂$

Here’s what students ask on this topic:

What is the cross product of two vectors?

The cross product of two vectors is a vector that is perpendicular to both of the original vectors. It is calculated using a matrix with the unit vectors i, j, k in the first row, followed by the components of the vectors. The resulting vector is found by computing the determinant of this matrix. The cross product is useful in physics and engineering to find a vector orthogonal to a plane defined by two vectors.

How do you calculate the cross product of two vectors?

To calculate the cross product of two vectors, follow these steps: 1) Set up a matrix with the unit vectors i, j, k in the first row. 2) Place the components of the first vector in the second row and the components of the second vector in the third row. 3) Compute the determinant of this matrix by multiplying and subtracting the appropriate elements. The resulting vector is the cross product, which can be expressed in component form.

What is the significance of the cross product in physics?

In physics, the cross product is significant because it provides a way to find a vector that is perpendicular to two given vectors. This is particularly useful in applications such as calculating torque, where the direction of the torque vector is perpendicular to the plane formed by the force and the lever arm. The magnitude of the cross product also represents the area of the parallelogram formed by the two vectors.

What are the properties of the cross product?

The cross product has several important properties: 1) It is anti-commutative, meaning $a\times b=-b\times a$. 2) It is distributive over vector addition, meaning $a\times (b+c)=a\times b+a\times c$. 3) The cross product of parallel vectors is zero. 4) The magnitude of the cross product is equal to the area of the parallelogram formed by the two vectors.

How do you determine the direction of the cross product vector?

The direction of the cross product vector is determined using the right-hand rule. Point the index finger of your right hand in the direction of the first vector (u) and your middle finger in the direction of the second vector (v). Your thumb will then point in the direction of the cross product (u × v). This ensures that the resulting vector is perpendicular to the plane formed by the original vectors.