Given u → ⋅ (v → × w →) = 3, find the following, i. v → ⋅ (u → × w →) ii. (u → × w →) ⋅ v → iii. v → ⋅ (w → × w →)

Question

I have one question that I don't understand in my homework. I'm not necessarily looking for the answer, but advice on HOW to solve this problem with steps so I can add it to my exam crib notes, which are due by Sunday night for prof review so I can use them to pass (I have a learning disability, so I need them, badly).

I've scoured the net for hours, read and re-read my textbook, watched countless videos, and I'm still stuck. Please help!

Answer 1

The scalar triple product is invariant under cyclic permutation, i.e., $$\tag1\vec a\cdot(\vec b\times \vec c)=\vec b\cdot(\vec c\times \vec a)=\vec c\cdot(\vec a\times \vec b).$$ The cross product is alternating, i.e., $$\tag2\vec a\times \vec b=-\vec b\times \vec a$$ and the dot product is symmetric, i.e., $$\tag3\vec a\cdot\vec b=\vec b\cdot \vec a.$$ Moreover, as a consequence of $(2)$, we have $$ \tag4\vec a\times \vec a=\vec0$$ and by linearity of the dot product $$\tag5 \vec a\cdot \vec 0=\vec 0\cdot \vec a=0.$$

Using this, $$\tag{i}\vec v\cdot(\vec u\times \vec w)\stackrel{(1)}= \vec u\cdot(\vec w\times \vec v)\stackrel{(2)}=-\vec u\cdot(\vec v\times \vec w)=-3$$ $$\tag{ii}(\vec u\times \vec w)\cdot \vec v\stackrel{(3)}=\vec v\cdot(\vec u\times \vec w)\stackrel{(\text i)}=-3$$ $$\tag{iii}\vec v\cdot(\vec w\times \vec w)\stackrel{(4)}=\vec v\cdot\vec 0\stackrel{(5)}=0$$