Vectors & 3D Geometry
Coplanarity; scalar triple product
MJMT_Full_Test_02
Grade 12
Question:
If $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+7\hat{j}+2\hat{k}$, $\vec{x}\cdot\vec{a}=0$ and $\vec{x}\cdot\vec{c}=0$ for some non-zero vector $\vec{x}$, then value of $\vec{a}\cdot(\vec{b}\times\vec{c})$ is
Step-by-Step Solution
Key Concept: If $\vec{x}\cdot\vec{a}=\vec{x}\cdot\vec{b}=\vec{x}\cdot\vec{c}=0$, then $\vec{x}$ is perpendicular to $\vec{a}$, $\vec{b}$, $\vec{c}$, which forces $\vec{a}$, $\vec{b}$, $\vec{c}$ to be coplanar.
$\vec{a}$, $\vec{b}$, $\vec{c}$ coplanar $\Rightarrow \vec{a}\cdot(\vec{b}\times\vec{c})=0$.
Correct Answer: 0