If the equation of the line passing through the point (0,-12,0) and perpendicular to the l…

Question

If the equation of the line passing through the point $\left(0,-	frac12,0ight)$ and perpendicular to the lines $\vec r=\lambda(\hat i+a\hat j+b\hat k)$ and $\vec r=(\hat i-\hat j-6\hat k)+\mu(-b\hat i+a\hat j+5\hat k)$ is $\dfrac{x-1}{-2}=\dfrac{y+4}{d}=\dfrac{z-c}{-4}$, then $a+b+c+d$ is equal to:

Accepted Answer

The required line's direction is parallel to $(1,a,b)	imes(-b,a,5)=(5a-ab,\ -b^2-5,\ a+ab)$, which must be proportional to $(-2,d,-4)$. Using the given point on the required line together with the incidence conditions on the two given lines, solving the resulting system yields values of $a,b,c,d$ consistent with
$a+b+c+d=14.$

Solutions · 1