JEE Mains · Maths · STD 12 - 10. vector algebra
Let \(\hat{u}\) and \(\hat{v}\) be unit vectors inclined at an acute angle such that \(|\hat{u}\times\hat{v}|=\dfrac{\sqrt{3}}{2}\). If \(\vec{A}=\lambda\hat{u}+\hat{v}+(\hat{u}\times\hat{v})\), then \(\lambda\) is equal to:
- A \(\dfrac{4}{3}(\vec{A}\cdot\hat{u})-\dfrac{2}{3}(\vec{A}\cdot\hat{v})\)
- B \(\dfrac{2}{3}(\vec{A}\cdot\hat{u})-\dfrac{1}{3}(\vec{A}\cdot\hat{v})\)
- C \(\dfrac{4}{3}(\vec{A}\cdot\hat{u})+\dfrac{2}{3}(\vec{A}\cdot\hat{v})\)
- D \((\vec{A}\cdot\hat{u})-\dfrac{1}{2}(\vec{A}\cdot\hat{v})\)
Answer & Solution
Correct Answer
(A) \(\dfrac{4}{3}(\vec{A}\cdot\hat{u})-\dfrac{2}{3}(\vec{A}\cdot\hat{v})\)
Step-by-step Solution
Detailed explanation
Given \(|\hat{u}\times\hat{v}| = \dfrac{\sqrt{3}}{2}\) and \(\hat{u}, \hat{v}\) are unit vectors. \(\sin\theta = \dfrac{\sqrt{3}}{2}\) Since the angle \(\theta\) is acute, \(\theta = \dfrac{\pi}{3}\). \(\hat{u}\cdot\hat{v} = \cos\dfrac{\pi}{3} = \dfrac{1}{2}\) Given…
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