Cross Product - Online Tutor, Practice Problems & Exam Prep

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 cross down and up method to find each component, recognizing the pattern of multiplying unlike components. The final vector can be expressed in component form, illustrating the relationship between the vectors. Understanding this process is essential for grasping vector operations in multivariable contexts.

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 multiply 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. 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. It would look something like that.

Now, as 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. I can see that we have vector u, which is 201 and vector v, which is 0−12 .

Our first step with the cross product should be to write a matrix for the i, j, and k components for each vector. In this matrix, the i, j, and k go on top. In the second row here, we're going to have our vector u. We write its components: the i component is 2, the j component is 0, and then the k component is 1, making up vector u. Below that, we write the components for vector v, which are 0, negative 1, 2.

Our second step is going to be to repeat the i and j columns outside of the matrix. So, I'm going to repeat the i column, which is 20, and also the j column, which is 0 negative 1. Then, our third step is to write uv − vu for each component. This is the pattern for each component. The cross product operation is then performed by using this matrix and multiplying down then up for each product. This creates a cross pattern, which is emblematic of a cross product.

Then, we calculate. For i, we have 0 times 2 which is 0, minus negative one times 1 which is positive 1. So the x component is 1. Then, for j, we have 1 times 0 which is 0, minus 2 times 2 which is 4, giving us -4 for the y component. Lastly, for k, we have 2 times negative one which is negative 2, minus 0 times 0 which is 0, giving -2 for the z component. Concatenating these, we get 1 i − 4 j − 2 k as our vector result in component form: 1, -4, -2.

While this process is tedious, by following the described matrix strategy, we identified and applied the unlike components being multiplied each time. This pattern helps with accurately computing the cross product, demonstrating that you just subtract these results to find the vector in question. I hope you found this video helpful. Thank you 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 that includes the unit vectors i, j, k in the first row, followed by the components of the two vectors. The resulting vector's components are found by using the cross down and up method, which involves multiplying and subtracting the unlike components of the vectors. The final vector can be expressed in component form, showing the relationship between the original 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, followed by the components of the two vectors. 2) Use the cross down and up method to find each component of the resulting vector. This involves multiplying the unlike components of the vectors and subtracting the results. 3) The final vector is expressed in component form, showing the x, y, and z components of the new vector.

What is the significance of the cross product in vector operations?

The cross product is significant in vector operations because it results in a vector that is perpendicular to the original vectors. This property is useful in various applications, such as finding the normal vector to a plane, calculating torque in physics, and determining the area of a parallelogram formed by two vectors. Understanding the cross product helps in grasping more complex multivariable concepts and operations.

What are the steps to find the cross product using the matrix method?

To find the cross product using the matrix method, follow these steps: 1) Write a matrix with the unit vectors i, j, k in the first row. 2) In the second row, write the components of the first vector. 3) In the third row, write the components of the second vector. 4) Repeat the i and j columns outside the matrix. 5) Use the cross down and up method to find each component of the resulting vector by multiplying and subtracting the unlike components. 6) Combine the components to form the final vector.

Why is the cross product of two vectors perpendicular to the original vectors?

The cross product of two vectors is perpendicular to the original vectors due to the geometric properties of the vectors involved. When you perform the cross product operation, you are essentially finding a vector that lies in a plane orthogonal to the plane formed by the original vectors. This perpendicularity is a fundamental characteristic of the cross product and is useful in various applications, such as determining the orientation of surfaces and calculating torque.