How do you find vector a and b given the cross product of a x b?

Question

Vocaloid · Answer

@hero @angle would you mind taking a look at this one? I'm not sure tbh

nuts · Answer

you need the dot product and cross product, it's a classic intro multivariate calc problem

nuts · Answer

and iirc there's two possible solutions

Liam · Answer

Suppose$$a^ \rightarrow$$ and $$b^ \rightarrow$$ are vectors such that $$a^ \rightarrow \times b^ \rightarrow$$ = (1,0,2). What is $$b^ \rightarrow \times (-2a^ \rightarrow )$$

Liam · Answer

Sorry, I'm not sure why it turned out like that.

nuts · Answer

what are the properties of the cross product that you know of?

Liam · Answer

We didn't learn the properties of the cross product. We were just taught how to do it.

nuts · Answer

you know of the a x b = a*b sin theta right?

Liam · Answer

Ah yes, we did talk about that.

nuts · Answer

is that enough of a hint, or do i take it further?

Liam · Answer

So we have: 
(1,0,2) = a*bsin theta 
But what else could I do without an angle or anything?

nuts · Answer

ok first off, when you switch the order of the vectors in a cross product, what happens

Liam · Answer

Changes sign?

nuts · Answer

would that cancel out with the negative in the a?

Liam · Answer

The -2a part?

nuts · Answer

yeah

nuts · Answer

and think about what the 2 does

Liam · Answer

So the -2a becomes 2a and the 2 double the value of a

nuts · Answer

yup, but due to that formula i gave you double the value of a is twice the length (but the same angle) so you just multiply your answer by 2

Liam · Answer

So we multiply the cross product by 2? Sorry, I'm sorta confused.

nuts · Answer

yup
b x (-2 a) = -(-2 a) x b = 2a x b = 2(a x b)

Liam · Answer

(2,0,4) ?

nuts · Answer

yup

Liam · Answer

But how does that give me the value of vector a and b?

nuts · Answer

why do you want to know what a and b are

nuts · Answer

if I have 2 cars and I sell one, I have one car left. I don't need to know how much the cars cost or how much I sold them for.

nuts · Answer

another way to put it, a and b are not unique vectors. Let me come up with an analogy

if a+b=2, what is 2a+2b? we know it's 4.

What are a and b? there are an infinite number of possibilities, but we still get 4 if we multiply their sum by 2.

Liam · Answer

Ohh, right!! What am I thinking? I get it now. Tysm

nuts · Answer

yup, gl!