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

AI Thread Summary
The discussion focuses on proving the vector equation A x (B + C) = A x B + A x C using the properties of the cross product. Participants suggest defining each vector in component form and applying the cross product definition. There is a specific inquiry about using the sine function to determine if two vectors are parallel, given the vectors u and v. The original poster expresses uncertainty about the final steps of the proof. Clarity on the parallelism of the vectors is sought to advance the discussion.
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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.
 
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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 defintion of the cross product in each case.
 
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
 

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