Gretchen Schwartz

2022-07-06

Let say I have 2 vectors (1,0,0) and (0,2,0), and I want to find a third vector that is orthogonal to both of them. I can do a cross product and get (0,0,2). However, I know there are infinite vector in the following form $\left(0,0,x\right)$ where $x$$\in$$R$ that are orthogonal to the other two. My question is what is the difference between the orthogonal vector results from cross product and any other orthogonal vector? Why the cross product gives only 1 specific orthogonal vector? What is the significance of this vector? Thank you!

Expert

The vector you get by performing the cross product is the unique vector orthogonal to both of your original vectors that

1. has a length equal to the magnitude of the area of the parallelogram (actually rectangle in this case) with sides (1,0,0) and (0,2,0) and
2. forms a right-handed set with (1,0,0) and (0,2,0)

If you don't care about either of those two properties, then you could just choose any vector of the form $\left(0,0,c\right)$. But sometimes those properties are useful.

Joel French

Expert

If $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$ are two vectors then the vector $\stackrel{\to }{a}×\stackrel{\to }{b}$ is another vector perpendicular, or normal, the plane formed by $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$. As you rightly suspected, there are an infinite number of vectors normal to the plane formed by $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$. They are all of the form $\alpha \left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)$, where $\alpha$ is a scalar. Since all of these "point" to the same direction, it is customary to choose a vector that has unit magnitude. It is given by
$\stackrel{\to }{c}=\frac{\stackrel{\to }{a}×\stackrel{\to }{b}}{|\stackrel{\to }{a}×\stackrel{\to }{b}|},$
where $|\cdot |$ denotes the magnitude of a vector.

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