Hi MHF
This is my first post, I'm faaaairly certain this is in the right subforum, sorry if it isn't.
Anyway, I'm having difficulty with vector algebra, more specifically, proving one expression equals another
I have this question:
"Prove (a - b) x (a + b) = 2(a x b)"
Where a and b are both vectors and x= cross product
The problem is, I am getting confused over what rules I can use from traditional algebra (ie taking a and b as just variables) and the vector specific rules, (ie commutative laws, distributive laws, etc)
I have another one too:
" if (c-0.5a).a = (c-0.5b).b = 0 then, prove c-0.5(a+b) is perpendicular to a - b"
where . = dot product
(Sorry if that one is confusing, typing out 1/2 was a bit messy compared to just typing 0.5)
I know that vectors are perpendicular if they have a dot product of 0, but (again) I'm just quite confused about what rules I can use here.
Thanks very much
You can't simply treat vectors as variables (first off, variables are scalar). Also, cross products are anti-commutative, unlike regular numbers.
For the first question, you could try drawing two random vectors a and b, as well as a+b, a-b. The magnitude of the cross product of two vectors is the area of the parallelogram spanned by those vectors.
Edit: topsquark's solution is faster...I forgot cross products distribute over addition.
This one comes a bit harder. First let's take the top equation apart into two pieces.
Again
Keep these under your hat for a moment.
You must show that
To go from here: Expand out this last line and you will find and . Plug in the equations we showed above. Then simplify.
-Dan