Let a= i+2 j+ k, b=2 i+7 j+3 k.

Question

Let $\vec a=\hat i+2\hat j+\hat k$, $\vec b=2\hat i+7\hat j+3\hat k$. Let $L_1: \vec r=(-\hat i+2\hat j+\hat k)+\lambda\vec a$ and $L_2:\vec r=(\hat j+\hat k)+\mu\vec b$. If $L_3$ passes through the intersection point of $L_1,L_2$ and is parallel to $\vec a+\vec b$, then $L_3$ passes through the point:

Accepted Answer

Equating $L_1,L_2$ components: $\lambda-1=2\mu$, $2(\lambda+1)=1+7\mu$, $\lambda+1=1+3\mu$. Solving gives $\lambda=3,\ \mu=1$, so the intersection point is $(2,8,4)$.

$\vec a+\vec b = 3\hat i+9\hat j+4\hat k$, so $L_3: \vec r = 2\hat i+8\hat j+4\hat k+\alpha(3\hat i+9\hat j+4\hat k)$. At $\alpha=2$: $\vec r = 8\hat i+26\hat j+12\hat k$, i.e. the point $(8,26,12)$.

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