Vector perpendicular to three vectors

Question

                           u = i+2j+3k
                           v = -i+2j+k
                           w = 3i+j

find the value of p such that (u+pv) is perpendicular to w

Answer 1

Hint:

If two non-zero vectors $\vec{a} \text{ and } \vec{b} $ are perpendicular and then,

$$\vec{a} \cdot\vec{b} = 0 \iff \vec{a} \perp \vec{b} \text{ as } |a|, |b| \ne 0$$

Answer 2

From $(u+pv)\cdot w=0$ and $v\cdot w\neq0$ give$$p=-\frac{u\cdot w}{v\cdot w}$$ a try.