Question

For any three vectors a,b,c . ( a-b). (b-c) ×(c-a) is equal

Answer 1

Just use the fact that A⃗ ×B⃗ =−B⃗ ×A⃗ A→×B→=−B→×A→

Use distributive laws to get A⃗ ×(B⃗ +C⃗ )=A⃗ ×B⃗  + A⃗ ×C⃗ A→×(B→+C→)=A→×B→ + A→×C→

B⃗ ×(C⃗ +A⃗ )=B⃗ ×C⃗  + B⃗ ×A⃗ B→×(C→+A→)=B→×C→ + B→×A→

C⃗ ×(A⃗ +B⃗ )=C⃗ ×A⃗  + C⃗ ×B⃗ C→×(A→+B→)=C→×A→ + C→×B→

Now just add them and rearrange to get this :

(A×B+B×A)+(B×C+C×B)+(C×A+A×C)(A×B+B×A)+(B×C+C×B)+(C×A+A×C)

But A⃗ ×B⃗ =−(B⃗ ×A⃗ )A→×B→=−(B→×A→)

B⃗ ×C⃗ =−(C⃗ ×B⃗ )B→×C→=−(C→×B→)

C⃗ ×A⃗ =−(A⃗ ×C⃗ )C→×A→=−(A→×C→)

So you get 0 + 0 + 0 = 0

Tell me if you need further help.

Expand all the brackets like this.

You get 6 term in total.

For a term of kind A⃗ ×B⃗ A→×B→ there is a term of the form B⃗ ×A⃗ B→×A→

Which cancel out when added. There are 3 such pairs who are cyclic permutations of one another.