Vectors
Vectors
Allen Star Batch
Grade 12
Question:
If $\vec{\alpha}$ and $\vec{\beta}$ be two perpendicular unit vectors such that $\vec{x} = \vec{\beta} - (\vec{a} \times \vec{x})$, then $|\vec{x}|$ is equal to:
$1$
$\sqrt{2}$
$\frac{1}{\sqrt{2}}$
None of these
Step-by-Step Solution
Key Concept: Combined use of cross and dot products systematically eliminates unknowns.
Taking cross product of $\vec{a} \times \vec{x} = \vec{a} \times \vec{b}$ with $\vec{a}$ gives $\vec{a} \times \vec{x} = \vec{a} \times \vec{b} + \vec{x} - (\vec{a} \cdot \vec{x})\vec{a}$. Taking dot product with $\vec{a}$ yields $\vec{a} \cdot \vec{x} = 0$, so $2\vec{x} = \vec{b} - (\vec{a} \times \vec{b})$ giving $|\vec{x}| = \frac{1}{\sqrt{2}}$.
Correct Answer: 3