Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade 12
Question:
Let $OPQR$ is a tetrahedon such that O is origin and $\vec{p}, \vec{q}, \vec{r}$ are position vectors of P, Q, R respectively and $\alpha$ is the angle which OP makes with face PQR then:
$|\sin\alpha| = \frac{|\vec{p}\cdot\vec{q}\vec{r}|}{|\vec{q} \times \vec{r} + \vec{r} \times \vec{p} + \vec{p} \times \vec{q}| \cdot |\vec{p}|}$
$\sin^2\alpha - \cos^2\alpha = \sqrt{3}$
$\tan\alpha = \frac{|\vec{p}\vec{q}\vec{r}|}{|\vec{q} \times \vec{r} + \vec{r} \times \vec{p} + \vec{p} \times \vec{r}| \cdot |\vec{p}|}$
None of these
Step-by-Step Solution
Key Concept: The angle between a line and a plane equals $\pi/2$ minus the angle with the normal.
Any vector perpendicular to plane $PQR$ is given by $\vec{PQ} \times \vec{PR} = (\vec{q} - \vec{p}) \times (\vec{r} - \vec{p}) = \vec{q} \times \vec{r} + \vec{p} \times \vec{q} + \vec{r} \times \vec{p}$. If line $OP$ makes angle $\alpha$ with plane $PQR$, then it makes angle $\frac{\pi}{2} - \alpha$ with the normal to the plane.
Correct Answer: 1