Cross Product. How does this: `C = A x B` formula used correctly?

Question

I don't understand how this: C = A x B formula works, Does the "x" means multiply?

If the vector a = 5 (magnitude) is going into the screen, then b = 7 is going up... Why the Cross product goes right? how does this: C = A x B formula used correctly? Do I just simply multiply 7 to 5 and get 35?

https://i.sstatic.net/Xd2Gi.png

If $\mathbf a$ and $\mathbf b$ happen to be orthogonal, then their cross product will have magnitude $35$. But that's not the case if they're at any other angle to one another. — , Oct 06 '16 at 14:15
You can get the cross product symbol with \times, so C=A \times B renders as $C=A\times B$ — Ross Millikan, Oct 06 '16 at 14:16
I suggest you take a bit to peruse some of the past questions on math.SE. In particular, 1, 2, 3, and 4 should be a good start. — , Oct 06 '16 at 14:17

score 1 · Answer 1 · answered Oct 06 '16 at 14:18

The magnitude of $A \times B$ is $|A||B|\sin \theta$, where $\theta$ is the angle between the vectors. In your example the vectors are orthogonal, so the angle is $\frac \pi 2$ and the $\sin$ is $1$. If the vectors are not orthogonal the length of the cross product will not be the product of the lengths. Try $(1,0,0) \times (1,1,0)$. The lengths are $1, \sqrt 2$ but the cross product is $(0,0,1)$ with length $1$.

Cross Product. How does this: `C = A x B` formula used correctly?

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