Let $a$ and $b$ be three-dimensional vectors. Then $(5a+7b)×(-a+3b)=ka×b$ for some scalar $k$. Find $k$.
Uhhhh I don't know how to start. Am I supposed to multiply the first 2 equations like $(5a^2+8ab+21b^2)$? maybe not tho...
Please provide tips to get started!! Thank you in advance.
Yes, you expand the expression first. $$(5a+7b)×(-a+3b)=5a×-a+5a×3b+7b×-a+7b×3b$$ The cross product of any vector with a linear multiple of itself is zero: $$=5a×3b+7b×-a$$ The cross product is also antisymmetric: $$=5a×3b+a×7b=22a×b$$ Thus $k=22$.