Vectors
Dot and cross product conditions with plane
MJAT_TS7_P1
Grade 12
Question:
Let $\overrightarrow{OP}=(\alpha^{-1})\hat{i}+\hat{j}+\hat{k}$, $\overrightarrow{OQ}=\hat{i}+(\beta^{-1})\hat{j}+\hat{k}$, $\overrightarrow{OR}=\hat{i}+\hat{j}+\frac{1}{2}\hat{k}$ where $\alpha,\beta\in\mathbb{R}\setminus\{0\}$. If $(\overrightarrow{OP}\times\overrightarrow{OQ})\cdot\overrightarrow{OR}=0$ and the point $(\alpha,\beta,2)$ lies on $3x+3y-z+l=0$, then $l$ equals:
Step-by-Step Solution
Key Concept: $\overrightarrow{OP}\times\overrightarrow{OQ}=\det\begin{pmatrix}\hat{i}&\hat{j}&\hat{k}\\1/\alpha&1&1\\1&1/\beta&1\end{pmatrix}$. Set dot with $\overrightarrow{OR}$ to 0. Find $\alpha,\beta$, then use the plane condition.
$l=\mathbf{5}$.
Correct Answer: 5