Question

Given that w = ku + mv, prove algebraically that u X v * w = 0?

k and m are scalars, u, v, and w are vectors. * is dot product. X is cross.

Thanks

Answer 1

first off, I have to recall the properties of dot product and cross product we use to prove this problem.
To dot product: \(\vec a\bullet \vec b= \vec b\bullet\vec a\)
To cross product \((\vec a +\vec b)\otimes \vec c = \vec a\otimes \vec c +\vec b\otimes \vec c\)
To combination: \(\vec c \bullet (\vec a\otimes \vec b) = (\vec c\otimes \vec a)\bullet vec b\)
Now, apply
\((\vec u\otimes \vec v)\bullet \vec w = \vec w\bullet (\vec u \otimes \vec v)\)=\((\vec w\otimes \vec u )\bullet \vec v\)
Replace \(\vec w = k\vec u+m\vec v\)
\(( (k\vec u +m\vec v)\otimes \vec u))\bullet \vec v\)
\(=((k\vec u\otimes \vec u)+(m\vec v\otimes \vec u))\bullet \vec v\)
the frist term =0.(I think you know why)
so, it becomes
\((m\vec v\otimes \vec u)\bullet \vec v\)=\(\vec v \bullet(m\vec v\otimes \vec u)\)

\(= (\vec v\otimes m\vec v)\bullet \vec u\)
\(=0\bullet \vec u =0\)

Answer 2

Thank you so much :)