Question

Let a unit vector $\widehat{ OP }$ make angles $\alpha, \beta, \gamma$ with the positive directions of the co-ordinate axes $OX$, $OY , OZ$ respectively, where $\beta \in\left(0, \frac{\pi}{2}\right)$ If $\widehat{ OP }$ 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 ?

• A. $\alpha \in\left(0, \frac{\pi}{2}\right)$ and $\gamma \in\left(0, \frac{\pi}{2}\right)$
• B. $\alpha \in\left(\frac{\pi}{2}, \pi\right)$ and $\gamma \in\left(\frac{\pi}{2}, \pi\right)$
• C. $\alpha \in\left(\frac{\pi}{2}, \pi\right)$ and $\gamma \in\left(0, \frac{\pi}{2}\right)$
• D. $\alpha \in\left(0, \frac{\pi}{2}\right)$ annd $\gamma \in\left(\frac{\pi}{2}, \pi\right)$

Answer 1

Answer: (B)

$\alpha \in\left(\frac{\pi}{2}, \pi\right)$ and $\gamma \in\left(\frac{\pi}{2}, \pi\right)$

Answer 2

Equation of plane :-
∣∣​x−110​y−213​z−314​∣∣​=0
⇒[x−1]−4[y−2]+3[z−3]=0
⇒x−4y+3z=2
D.R's of normal of plane <1,−4,3>
D.C's of
⟨±26​1​,∓26​4​,±26​3​⟩
cosβ=26​4​
cosα=26​−1​2π​<α<π
cosγ=26​−3​2π​<γ<π