Vectors
Vectors
Allen Star Batch
Grade 12
Question:
If $\vec{a}$ is a unit vector and projection of $\vec{x}$ along $\vec{a}$ is 2 units and $(\vec{a} \times \vec{x}) \cdot \vec{b} = \vec{x}$, then $\vec{x}$ is given by:
$\frac{1}{2}[\vec{a} - \vec{b} + (\vec{a} \times \vec{b})]$
$\frac{1}{2}[2\vec{a} + \vec{b} + (\vec{a} \times \vec{b})]$
$[\vec{a} + (\vec{a} \times \vec{b})]$
None of these
Step-by-Step Solution
Key Concept: Use vector triple product expansion and dot product conditions to isolate the unknown vector.
From the given condition $(\vec{a} \times \vec{x}) + \vec{b} = \vec{x}$, apply cross product with $\vec{a}$: $\vec{a} \times (\vec{a} \times \vec{x}) + (\vec{a} \times \vec{b}) = \vec{a} \times \vec{x}$. Using the vector triple product formula and simplifying with the projection condition $\frac{(\vec{a} \cdot \vec{x})}{|\vec{a}|} = 2$ gives $\vec{a} \cdot \vec{x} = 2$. Solving the system yields $\vec{x} = \frac{1}{2}[2\vec{a} + \vec{b} + (\vec{a} \times \vec{b})]$.
Correct Answer: 2