Let a = i+2 j+ k and b = 2 i+ j- k.

Question

Let $\vec a = \hat i+2\hat j+\hat k$ and $\vec b = 2\hat i+\hat j-\hat k$. Let $\hat c$ be a unit vector in the plane of $\vec a$ and $\vec b$, perpendicular to $\vec a$. Then $\hat c$ is:

Accepted Answer

Let $\vec p = K(\vec a+\lambda\vec b)$ lie in the plane of $\vec a,\vec b$. For $\vec p\perp\vec a$: $\vec p\cdot\vec a=0 \Rightarrow \vec a\cdot\vec a+\lambda(\vec b\cdot\vec a)=0$.

$\vec a\cdot\vec a=6$, $\vec b\cdot\vec a = 2+2-1=3$: $6+3\lambda=0 \Rightarrow \lambda=-2$.

$$\vec p = \vec a-2\vec b = (1-4)\hat i+(2-2)\hat j+(1+2)\hat k = -3\hat i+3\hat k$$

$$\hat c = \pm\frac{-\hat i+\hat k}{\sqrt2}$$

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