Question

If \(\hat u,\hat v\) are unit vectors and \[ |\hat u\times \hat v|=\frac{\sqrt3}{2} \] and \[ \vec A=\lambda\hat u+\hat v+\hat u\times \hat v \] then find \(\lambda\). (Angle between \(\hat u\) and \(\hat v\) is acute)

• A. \(\lambda=\frac13\vec A\cdot\hat u-\frac13\vec A\cdot\hat v\)
• B. \(\lambda=\frac43\vec A\cdot\hat u-\frac23\vec A\cdot\hat v\)
• C. \(\lambda=\frac83\vec A\cdot\hat u-\frac23\vec A\cdot\hat v\)
• D. \(\lambda=\frac23\vec A\cdot\hat u-\frac43\vec A\cdot\hat v\)

Hint:

For unit vectors, \(|\vec a\times\vec b|=\sin\theta\) and \(\vec a\cdot\vec b=\cos\theta\).

Answer 1

The Correct Option is B

Concept: For unit vectors \[ |\hat u\times\hat v|=|\hat u||\hat v|\sin\theta=\sin\theta \] where \(\theta\) is the angle between them.
Step 1: Find the angle \[ |\hat u\times\hat v|=\frac{\sqrt3}{2} \] \[ \sin\theta=\frac{\sqrt3}{2} \] Since the angle is acute, \[ \theta=\frac{\pi}{3} \] Thus \[ \hat u\cdot\hat v=\cos\frac{\pi}{3}=\frac12 \]
Step 2: Given vector \[ \vec A=\lambda\hat u+\hat v+\hat u\times\hat v \]
Step 3: Dot with \(\hat u\) \[ \vec A\cdot\hat u=\lambda(\hat u\cdot\hat u)+\hat v\cdot\hat u+(\hat u\times\hat v)\cdot\hat u \] \[ \vec A\cdot\hat u=\lambda+\frac12 \] \[ 2(\vec A\cdot\hat u)=2\lambda+1 \quad ...(1) \]
Step 4: Dot with \(\hat v\) \[ \vec A\cdot\hat v=\lambda(\hat u\cdot\hat v)+\hat v\cdot\hat v+(\hat u\times\hat v)\cdot\hat v \] \[ \vec A\cdot\hat v=\frac{\lambda}{2}+1 \] \[ \vec A\cdot\hat v-1=\frac{\lambda}{2} \quad ...(2) \]
Step 5: Eliminate \(\lambda\) From (1) and (2), \[ 2\vec A\cdot\hat u-2\lambda=\vec A\cdot\hat v-\frac{\lambda}{2} \] Solving, \[ \lambda=\frac43\,\vec A\cdot\hat u-\frac23\,\vec A\cdot\hat v \]