Ask Me Help Desk - Cross product

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cross product

I am not sure how I should explain the following problems:

If you take any two perpendicular unit vectors X and Y, then

1. X\[Cross]Y is perpendicular to both X and Y

2. X\[Cross]Y is also a unit vector

3. If X and Y are unit vectors, but X is not perpendicular to Y, then can X\[Cross]Y be a unit vector? Why not?

Recall that the definition of the cross product is:

$<br /> \vec a \times \vec b = |a| |b| sin \theta \ \hat n<br />$

where $\theta$ is the smaller angle between $\vec a$ and $\vec b$ , and $\hat n$ is a unit vector perpendicular to the plane containing $\vec a$ and $\vec b$ . If the magnitude of $\vec a$ and $\vec b$ are both 1, and $\theta$ is something other than 90 degrees, can the magnitude of the cross product ever be equal to 1?

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