Let a line L passing through the point (1,1,1) be perpendicular to two given vectors whose…

Question

Let a line $L$ passing through the point $(1,1,1)$ be perpendicular to two given vectors whose cross product yields direction ratios $(2,-3,2)$. If $P(a,b,c)$ is the foot of perpendicular from the origin on the line $L$, then the value of $34(a+b+c)$ is:

Accepted Answer

$L:\ (1,1,1)+\lambda(2,-3,2)$, general point $M=(2\lambda+1,\ -3\lambda+1,\ 2\lambda+1)$.

Foot of perpendicular from origin: $\vec{OM}\cdot(2,-3,2)=0$:
$$2(2\lambda+1)-3(-3\lambda+1)+2(2\lambda+1)=0 \implies 17\lambda+1=0 \implies \lambda=-\frac{1}{17}.$$
$$M=\left(\frac{15}{17},\ \frac{20}{17},\ \frac{15}{17}\right)=(a,b,c).$$
$$34(a+b+c)=34\cdot\frac{50}{17}=100.$$

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