Vectors
Vectors
Allen Star Batch
Grade 12
Question:
If $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are any four vectors, then $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d})$ is a vector:
Perpendicular to $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$
Along the line of intersection of two planes, one containing $\vec{a}, \vec{b}$ other containing $\vec{c}, \vec{d}$
Equally inclined to both $\vec{a} \times \vec{b}$ and $\vec{c} \times \vec{d}$
None of these
Step-by-Step Solution
Key Concept: The direction of intersection of two planes is perpendicular to both plane normals, found via cross product of the normals.
The line of intersection of two planes is perpendicular to both normal vectors $\vec{n}_1 = \vec{a} \times \vec{b}$ and $\vec{n}_2 = \vec{c} \times \vec{d}$. The direction of the line of intersection is given by $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d})$. Find a specific point by solving both plane equations simultaneously, then write the parametric equation using this point and the direction vector.
Correct Answer: 2,3