1. ## Working with vectors

I know how to compute scalar and cross products but I'm not very familiar with what other rules applies to vectors.

For example If I have the vectors u and v and the following equation $u x (u+v) - v x (3u+v)$ and I know that u x v = {3,1,-1}
how do they arrive at 4uxv when simplifying the equation? Also in what order should you do the calculations, does cross product come before dot and addition/subtraction?

2. ## Re: Working with vectors

Originally Posted by dipsy34
I know how to compute scalar and cross products but I'm not very familiar with what other rules applies to vectors. For example If I have the vectors u and v and the following equation u x v(u+v) - v(3u+v) and I know that u x v = {3,1,-1}
how do they arrive at 4uxv when simplifying the equation? Also in what order should you do the calculations, does cross product come before dot and addition/subtraction?
It appears that part of your confusion may be use of notation. For example $v(3u+v)$ has no real meaning.
If it were $v\times (3u+v)$ then fine. That equals $3(v\times u)+(v\times v)=3(v\times u)$ because $(v\times v)=0~.$

Do you understand $u\times v$ is a vector, while $u\cdot v$ is a scalar.
Thus $(u\cdot v)\times w$ is meaningless. Whereas $u\times(v\times w)=(u\cdot w)v-(u\cdot v)w.$

You can repost your question with corrections.

3. ## Re: Working with vectors

Fixed the problem to the correct one.

4. ## Re: Working with vectors

Originally Posted by dipsy34
I know how to compute scalar and cross products but I'm not very familiar with what other rules applies to vectors. For example If I have the vectors u and v and the following equation $u \times (u+v) - v \times (3u+v)$ and I know that u \times v = {3,1,-1} how do they arrive at 4uxv when simplifying the equation?
$u \times (u+v)=(u\times u)+(u\times v)=(u\times v) ~$ and $~ - v \times (3u+v)=(- v \times 3u)=3(u\times v)$
Recall that $- v \times u= u\times v~.$

5. ## Re: Working with vectors

when using latex, use \times for the "x" symbol, instead of the letter x, which displays differently.

6. ## Re: Working with vectors

Originally Posted by Plato
$~ - v \times (3u+v)=(- v \times 3u)$
This is the part I'm struggling with atm, the others I follow, what are the rules for crossing in the -v in to the parenthesis?

7. ## Re: Working with vectors

the cross-product distributes over addition, so

a x (b + c) = (a x b) + (a x c).

here, your "a" is -v, the "b" is 3u, and the "c" is v, so we have:

-v x (3u + v) = (-v x 3u) + (-v x v)

the cross-product is also compatible with scalar multiplication:

(ra) x b = a x (rb) = r(a x b).

in particular, for r = -1, and a = b = v:

-v x v = (-1)v x v = (-1)(v x v) = (-1)(0) = 0.

thus

(-v x 3u) + (-v x v) = (-v x 3u) + 0 = -v x 3u