Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade 12
Question:
If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar, mutually perpendicular unit vectors, then $[\vec{a} \vec{q}] \vec{a} + [\vec{b} \vec{q}] \vec{b} + [\vec{c} \vec{q}] \vec{c}$ is equal to:
\vec{a} + \vec{b} + \vec{c}
\vec{p} \times \vec{q}
\vec{p} + \vec{q}
\vec{p} \times \vec{q}
Step-by-Step Solution
Key Concept: Projection of a cross product in a given direction uses scalar triple product properties.
The projection of $\vec{b} \times \vec{c}$ in the direction of $\vec{a}$ is computed using the dot product. The scalar projection equals $\frac{(\vec{b} \times \vec{c}) \cdot \vec{a}}{|\vec{a}|}$, which by properties of scalar triple product equals $\vec{a} \cdot (\vec{b} \times \vec{c})$. Therefore the given vector is $\vec{b} \times \vec{c}$.
Correct Answer: 4