Let a = 3 i+ j-2 k, b = 4 i+ j+7 k, and c = i-3 j+4 k.

Question

Let $\vec a = 3\hat i+\hat j-2\hat k$, $\vec b = 4\hat i+\hat j+7\hat k$, and $\vec c = \hat i-3\hat j+4\hat k$. If a vector $\vec p$ satisfies $\vec p	imes\vec b = \vec c	imes\vec b$ and $\vec p\cdot\vec a = 0$, then $\vec p\cdot(\hat i-\hat j-\hat k)$ is equal to

Accepted Answer

$(\vec p-\vec c)\times\vec b = \vec 0 \Rightarrow \vec p = \vec c+\lambda\vec b$.

$\vec p\cdot\vec a = 0$: $\vec c\cdot\vec a+\lambda(\vec b\cdot\vec a)=0$.

$\vec c\cdot\vec a = 3-3-8=-8$; $\vec b\cdot\vec a=12+1-14=-1$: $-8-\lambda=0 \Rightarrow \lambda=-8$.

$\vec p = \vec c-8\vec b = (1-32,\,-3-8,\,4-56) = (-31,-11,-52)$.

$$\vec p\cdot(\hat i-\hat j-\hat k) = -31+11+52 = 32$$

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