Question

for any three vector a, b, c prove that a(b+c)+b(c+a)+c(a+b)
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Answer 1

Answer:

follow this images please

Explanation:

Just use the fact that A⃗ ×B⃗ =−B⃗ ×A⃗

Use distributive laws to get A⃗ ×(B⃗ +C⃗ )=A⃗ ×B⃗ + A⃗ ×C⃗

B⃗ ×(C⃗ +A⃗ )=B⃗ ×C⃗ + B⃗ ×A⃗

C⃗ ×(A⃗ +B⃗ )=C⃗ ×A⃗ + C⃗ ×B⃗

Now just add them and rearrange to get this :

(A×B+B×A)+(B×C+C×B)+(C×A+A×C)

But A⃗ ×B⃗ =−(B⃗ ×A⃗ )

B⃗ ×C⃗ =−(C⃗ ×B⃗ )

C⃗ ×A⃗ =−(A⃗ ×C⃗ )

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 there is a term of the form B⃗ ×A⃗

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

I hope you get the solution.