Let a plane P contain two lines r= i+( i+ j), R, and r=- j+( j- k), R.

Question

Let a plane $P$ contain two lines $\vec r=\hat i+\lambda(\hat i+\hat j)$, $\lambda\in\mathbb R$, and $\vec r=-\hat j+\mu(\hat j-\hat k)$, $\mu\in\mathbb R$. If $Q(\alpha,\beta,\gamma)$ is the foot of the perpendicular drawn from the point $M(1,0,1)$ to $P$, then $3(\alpha+\beta+\gamma)$ equals \_\_\_\_\_\_\_.

Answer: 5

Accepted Answer

Normal $=(1,1,0)\times(0,1,-1)=(-1,1,1)$. Plane through $(1,0,0)$: $-(x-1)+y+z=0\implies x-y-z=1$.
Foot of perpendicular from $M(1,0,1)$: value $=1-0-1-1=-1$, $t=\tfrac{1}{3}$,
$$Q=M+\tfrac13(1,-1,-1)=\left(\tfrac43,-\tfrac13,\tfrac23\right)\implies \alpha+\beta+\gamma=\tfrac53\implies 3(\alpha+\beta+\gamma)=5.$$

Solutions · 1