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# Vector triple product expansion (very optional)

A shortcut for having to evaluate the cross product of three vectors. Created by Sal Khan.

## Want to join the conversation?

• At , you were doing the component j, should it be negative, not positive??
• In short, no. The sign swap business that happened with the first j was due to the fact that he is computing the determinant using the method of cofactors. So it's not something special about j, but rather a step in properly computing the determinant. So, if you properly compute the determinant of the second (big) matrix to get a x b x c, you'll get the result shown at .

Try it... it's a fun exercise. It's much more satisfying to do it yourself rather than watch someone else "smarter than you" do it on a blackboard. Once you've done it, then you're the smart one.
• Two cross products gives us a scalar! How is it possible?
• It is quite natural to get confused and view it as a scalar because our mind has got into this habit of viewing any dot product as a scalar. But if you look carefully and analyse, you will realise that it is not a scalar. It is difference of two vectors scaled by dot products. I.e
b.(a.c) - c.(a.b). ---- (1)

Suppose, I call the dot products a.c as m and a.b as n. (Note here m and n are scalars)
Then (1) can be written as
bm - cn
(Vector b scaled by m minus vector c scaled by n )
So it turns out to be difference of two scaled vectors and not a scalar as you view it to be. I hope this answer helped. :)
• how can i simply remember a x (b x c)?
• Where does Sal do videos on determinants? I cannot find it in this playlist.
• When Sal says multiply b and c by the dot products of a and c and a and b at (b(a dot c) - c(a dot b)) what does he mean by multiply?
• Scalar multiplication of the vector b with the scalar (a dot c) and so on
• Is vector multiplication associative? For instance, in the video, does it matter if it's a X (b X c), (a X b) X c, or a X b X c, or are they all equal?
• Unfortunately it is not. If you want to, just use some simple vectors and try it yourself. First calculate b x c and then a x the result. And then calculate a x b and take the result x c.
• How to find product of 3 expressions in optional math?
• If a x (b x c) = b(a⋅c) - c(a⋅b), then (b x c) x a = ? and (a x b) x c = ?
I know that cross products are neither commutative nor associative.
• You're right that it isn't commutative, but the good news is that it is what we call anti-commutative. That is, a x b = -(b x a). You can plug that into the formula and see it for yourself, or just use the right hand rule and the proof from two videos ago to see that b x a has the same magnitude and opposite direction as a x b.

Using that and the formula from this video, you can evaluate the two expressions you are interested in.