Let hat{u} and hat{v} be unit vectors incline | vedclass

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(A) Given $\vec{A} = \lambda \hat{u} + \hat{v} + (\hat{u} 	imes \hat{v})$.Since $\hat{u}$ and $\hat{v}$ are unit vectors, $|\hat{u} 	imes \hat{v}| =

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$\frac{4}{3}(\vec{A} \cdot \hat{u}) - \frac{2}{3}(\vec{A} \cdot \hat{v})$

Vedclass · Answer

(A) Given $\vec{A} = \lambda \hat{u} + \hat{v} + (\hat{u} 	imes \hat{v})$.Since $\hat{u}$ and $\hat{v}$ are unit vectors, $|\hat{u} 	imes \hat{v}| = |\hat{u}||\hat{v}| \sin 	heta = \sin 	heta = \frac{\sqrt{3}}{2}$.Since $	heta$ is acute, $	heta = 60^\circ = \frac{\pi}{3}$.Thus, $\hat{u} \cdot \hat{v} = \cos 60^\circ = \frac{1}{2}$.Taking the dot product of $\vec{A}$ with $\hat{u}$: $\vec{A} \cdot \hat{u} = \lambda(\hat{u} \cdot \hat{u}) + (\hat{v} \cdot \hat{u}) + ((\hat{u} 	imes \hat{v}) \cdot \hat{u}) = \lambda + \frac{1}{2} + 0 = \lambda + \frac{1}{2}$.Taking the dot product of $\vec{A}$ with $\hat{v}$: $\vec{A} \cdot \hat{v} = \lambda(\hat{u} \cdot \hat{v}) + (\hat{v} \cdot \hat{v}) + ((\hat{u} 	imes \hat{v}) \cdot \hat{v}) = \frac{\lambda}{2} + 1 + 0 = \frac{\lambda}{2} + 1$.Now, evaluate option $A$: $\frac{4}{3}(\vec{A} \cdot \hat{u}) - \frac{2}{3}(\vec{A} \cdot \hat{v}) = \frac{4}{3}(\lambda + \frac{1}{2}) - \frac{2}{3}(\frac{\lambda}{2} + 1) = \frac{4\lambda + 2 - \lambda - 2}{3} = \frac{3\lambda}{3} = \lambda$.

Let $\hat{u}$ and $\hat{v}$ be unit vectors inclined at an acute angle such that $|\hat{u} \times \hat{v}| = \frac{\sqrt{3}}{2}$. If $\vec{A} = \lambda \hat{u} + \hat{v} + (\hat{u} \times \hat{v})$, then $\lambda$ is equal to:

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