Vector relations to be proved?

Question

Using defination of Scalar and Vector product for vectors a,b,c
Given a X b = c, b X c= a, c X a= b (where X represents cross product)
Prove b.a = c.a = b.c = 0
and Also prove a.a = c.c & b.b=1 (where. represents dot or scalar product)

Answer 1

I agree with cw's answer. He unlocked a key piece of information for the first part.

I can probably chime in on the second part:

Ok, so we know that the a, b, c vectors are all at right angles with one another as cw explained. The formulas for the magnitude of the cross products would be as follows.
|a|*|b|*sin(90 deg) = |c|
|b|*|c|*sin(90 deg) = |a|
|c|*|a|*sin(90 deg) = |b|

(1) |a|*|b| = |c|
(2) |b|*|c| = |a|
(3) |c|*|a| = |b|

Ok, so lets substitute equation (1) into (2).
|a|*|b|^2 = |a|
|b|^2 = 1
|b| = 1

So we know that the magnitude of vector b (represented as |b|) is equal to one. With that you can prove that b.b =1. One objective out of the way.

You can also work with equations (1)-(3) to find that a.a = c.c via the same methodology. In fact, you should find that a.a =1 and c.c =1 as well.

That should get you pointed in the right direction for the second part.

Answer 2

This isn't a proof, because I did that already when I was in school.
But here's the idea. First you need to realize what you have been given.

A cross product of two vectors gives you the vector that is normal to the plane created by the initial vectors. So you are told that the vectors a,b,c are all orthogonal. If two vectors are orthogonal then the dot product is zero. So now you just have to do the math proof part.

The last one seems to be missing some information about the vectors. Without spending more time than I want, I can't comment on the last.