Let the lines L_1: r=( i+2 j+3 k), R, and L_2: r=( i+3 j+ k)+( i+ j+5 k), R, intersect at …

Question

Let the lines $L_1:\ \vec r=\lambda(\hat i+2\hat j+3\hat k)$, $\lambda\in\mathbb R$, and $L_2:\ \vec r=(\hat i+3\hat j+\hat k)+\mu(\hat i+\hat j+5\hat k)$, $\mu\in\mathbb R$, intersect at the point $S$. If a plane $ax+by-z+d=0$ passes through $S$ and is parallel to both the lines $L_1$ and $L_2$, then the value of $a+b+d$ is equal to:

Answer: 5

Accepted Answer

Setting the two parametrizations equal: $\lambda=1+\mu,\ 2\lambda=3+\mu,\ 3\lambda=1+5\mu$ gives $\mu=1,\ \lambda=2$.
$S=(2,4,6).$
Normal $(a,b,-1)$ perpendicular to both directions:
$$a+2b-3=0,\qquad a+b-5=0 \implies b=-2,\ a=7.$$
Plane: $7x-2y-z+d=0$ through $S$: $14-8-6+d=0\Rightarrow d=0$.
$a+b+d=7-2+0=5.$

Solutions · 1