3D Geometry
Direction cosines and angle with coordinate axes
nta_pyq_2023_jan
Grade 12
Question:
Let a unit vector $\overrightarrow{OP}$ make angles $\alpha, \beta, \gamma$ with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where $\beta \in \left(0, \frac{\pi}{2}\right)$. $\overrightarrow{OP}$ is perpendicular to the plane through points $(1,2,3)$, $(2,3,4)$ and $(1,5,7)$. Then which one of the following is true?
$\alpha \in \left(\frac{\pi}{2}, \pi\right)$ and $\gamma \in \left(\frac{\pi}{2}, \pi\right)$
$\alpha \in \left(0, \frac{\pi}{2}\right)$ and $\gamma \in \left(0, \frac{\pi}{2}\right)$
$\alpha \in \left(\frac{\pi}{2}, \pi\right)$ and $\gamma \in \left(0, \frac{\pi}{2}\right)$
$\alpha \in \left(0, \frac{\pi}{2}\right)$ and $\gamma \in \left(\frac{\pi}{2}, \pi\right)$
Step-by-Step Solution
Key Concept: Find normal to plane through 3 points; the normal gives direction cosines. Then determine quadrant of $\alpha$ and $\gamma$.
Normal to plane = $\vec{n_1}\times\vec{n_2}$ where rows are differences of points. Normal direction $(-1,7,11)$... wait: plane equation gives DRs $\langle -1,4,-3 \rangle$ (from solution page: $\cos\alpha = -1/\sqrt{26} < 0 \Rightarrow \alpha \in (\pi/2,\pi)$; $\cos\gamma = -3/\sqrt{26} < 0 \Rightarrow \gamma \in (\pi/2,\pi)$). Answer: (1)
Correct Answer: $\alpha \in \left(\frac{\pi}{2}, \pi\right)$ and $\gamma \in \left(\frac{\pi}{2}, \pi\right)$