Question

If a plane, that contains the line \(\overrightarrow{\mathrm{r}}=(1,-1,1)+\lambda(2,3,-6), \lambda \in \mathbf{R}\), passes through the point \((2,0,2)\) and is at a distance \(\mathrm{d}\) from the origin, then the value of \(73 \mathrm{~d}^{2}\) is ________.

Hint:

The normal is the cross product of the line direction and the vector from a point on the line to (2, 0, 2); then use the point-to-plane distance from the origin.

Answer 1

Correct Answer: 128

Point on the line: \(A(1,-1,1)\), direction \(\vec b=(2,3,-6)\). Given point \(C(2,0,2)\), so \(\vec{AC}=(1,1,1)\).
Normal \(\vec n=\vec b\times\vec{AC}=(3\cdot1-(-6)\cdot1,\ (-6)\cdot1-2\cdot1,\ 2\cdot1-3\cdot1)=(9,-8,-1)\).
Plane: \(9(x-1)-8(y+1)-(z-1)=0\), i.e. \(9x-8y-z-16=0\) (check at C: 18-0-2-16=0).
Distance from the origin: \(d=\frac{16}{\sqrt{81+64+1}}=\frac{16}{\sqrt{146}}\), so \(d^2=\frac{256}{146}=\frac{128}{73}\).
\(73d^2=128\).
Answer: 128