Vector Cross Product

This lesson will give you some help on finding the cross product of a pair of three-dimensional vectors. This is something which I saw in one of the courses of the three-semester General Physics series, and also again in Calculus III. In the remainder of my degree work I haven't used it much, so it's easy to get rusty on. However, I anticipate that you'll see at least one question on the Fundamentals of Engineering exam which asks for the cross product and potentially another that uses it, and it's a fairly quick and easy question to answer...if you remember how! So watch this video and then do a couple practice problems to brush up before your exam.

As you might notice if you've watched the previous video (and I mention it) the cross product is basically the determinant of a 3x3 matrix, except the top row is the <i, j, k> "vector" and the second and third rows are the coefficients of A and B respectively. So if you can do a 3x3 matrix determinant problem, you can do this.

The vector which results from taking a cross product is orthogonal or perpendicular to the plane formed by the vectors you are multiplying, and it's basically a mathematical calculation of the principle illustrated by the right-hand rule. If someone were to ask you to find a vector orthogonal or perpendicular to two vectors (on an exam, in your career, late at night at the local pub...) you can take the cross product of the two and arrive at an answer.

Cross products have applications in torque and angular momentum, so if you're a Mechanical student you might not be as rusty as I was. I'm a Civil, we just play in the dirt, right?

Tip: Remember, don't mix up your signs when you go to add up the parts of the determinant. Start at the top-left corner of the matrix. The products of the lines which run "northwest" to "southeast" are added; the ones which run "southwest" to "northeast" are subtracted.

Tip: Don't forget that the i-j-k row across the top is part of your matrix and you have to carry these along. Then when you sum them up, combine like terms. It follows that your result should always be another vector (in terms of i, j, and k), unlike the determinant of a purely-numerical matrix which gives you a scalar (for example, -42.) This might be a way to eliminate answers from your FE multiple-choice exam. If one or more of the possible answers is a scalar, you can eliminate those choices. Even if you can't work the problem to completion, at least by understanding the principle you've eliminated some possibilities and increased your chances of guessing correctly.

PRACTICE PROBLEMS: Find the cross product AxB of the following vectors:

a) A = <3i + 2j + 6k>, B = <7i + 3j + 4k>
b) A = <i + j + k>, B = <12i + 2j + 3k>
c) A = <9i - 3j + 4k>, B = <-i + 2j -7k>

If you feel like you need more practice you can make up your own practice problems. Dream up any combination of vectors you want and check your answers on WolframAlpha's cross product calculator.

SOLUTIONS: Highlight the text with your mouse to reveal the answers.

a) AxB = <-10i + 30j - 5k>
b) AxB = <i + 9j - 10k>
c) AxB = <13i + 59j + 15k>