Let a unit vector OP make angles ,, with the positive directions of the coordinate axes OX…

Question

Let a unit vector $\hat{OP}$ make angles $\alpha,\beta,\gamma$ with the positive directions of the coordinate axes $OX,OY,OZ$ respectively, where $\beta\in\left(0,\dfrac{\pi}{2}\right)$. If $\hat{OP}$ is perpendicular to the plane through the points $(1,2,3),\ (2,3,4),\ (1,5,7)$, then which one of the following is true?

Accepted Answer

Vectors $(1,1,1)$ and $(0,3,4)$ in the plane; normal $=(1,1,1)\times(0,3,4)=(1,-4,3)$.

Direction cosines $\propto\left(\dfrac{1}{\sqrt{26}},\dfrac{-4}{\sqrt{26}},\dfrac{3}{\sqrt{26}}\right)$ or the negative. Taking $\cos\beta>0$ (given $\beta\in(0,\pi/2)$) forces $\cos\alpha=\dfrac{-1}{\sqrt{26}}<0$ and $\cos\gamma=\dfrac{-3}{\sqrt{26}}<0$.
$$\alpha\in\left(\frac{\pi}{2},\pi\right),\qquad \gamma\in\left(\frac{\pi}{2},\pi\right).$$

Solutions · 1