Proving Cross Product Equation: A x (B + C) = A x B + A x C

[ex81]

Problem:
Prove A x (B + C) = A x C + A x B

Related equations:
Cross product

Attempts:
Not even remotely sure where to start with this, other than I need to use the cross product rule.

 

[bmxicle]

Try defining each vector $$(a_1, a_2, a_3), (b_1, b_2, b_3), (c_1, c_2, c_3)$$ and then using the definition of the cross product in each case.

 

[dilresh]

i have to ask that using Sinθ=u x v/|u||v|, these both vectors are 2-Dimensional u=-3i-5j,
v=4i+7j, u x v=-21i+20j ,|u|=√34,|v|=√65, Sinθ=-21i+20j/(√65√34)
question is check are they parallel?please can anyone help me for the last step