Vectors & 3D Geometry
Equally inclined vector and perpendicularity of diagonals
MJAT_TS6_P1
Grade 12
Question:
**Statement 1:** Let $OABC$ be a regular tetrahedron with $O$ at origin. If $P$ is a point at unit distance from origin equally inclined to $\overrightarrow{OA}$, $\overrightarrow{OB}$, $\overrightarrow{OC}$, then $p_1=\cos^2\alpha$ where $\alpha$ is the angle.
**Statement 2:** $A,B,C,D$ are four points with $AB=3$, $BC=7$, $CD=11$, $DA=9$. The angle between $AC$ and $BD$ is $\theta$ and $p_2=\cos\theta$.
The value of $[3p_1+2p_2]$ (GIF) is:
Step-by-Step Solution
Key Concept: **S1:** For regular tetrahedron, if $\hat{a}\cdot\hat{b}=\hat{b}\cdot\hat{c}=\hat{c}\cdot\hat{a}=-1/3$ (since all edges equal). Let $\hat{d}=x\hat{a}+y\hat{b}+z\hat{c}$: $\hat{d}\cdot\hat{a}=x+y(-1/3)+z(-1/3)=\cos\alpha$ for each. Symmetry $\Rightarrow x=y=z$: $3\cos^2\alpha=1\Rightarrow p_1=1/3$.
$3p_1=2$, $2p_2=0$. $[3p_1+2p_2]=\mathbf{2}$.
Correct Answer: 2