Question

Sol. [Key Concept: Dot $A$ with $u$ and with $v$; use $\theta = \pi/3$ (so $u \cdot \ v = \frac{1}{2}$) and solve the two relations for $\lambda$.]

$${|\widehat{u} \times \widehat{v}| = \frac{\sqrt{3}}{2} }{|\widehat{u}\|\widehat{v}|sin\theta = \frac{\sqrt{3}}{2}\ \therefore\theta = \frac{\pi}{3} }$$and $\widehat{u} \cdot \widehat{v} = |\widehat{u}||\widehat{v}|cos\frac{\pi}{3} = \frac{1}{2}$

$$\begin{array}{r} \overrightarrow{A} = \lambda\widehat{u} + \widehat{v} + \widehat{u} \times \widehat{v}\#(1) \end{array}$$

Dot with û

$${\overrightarrow{A} \cdot \widehat{u} = \lambda(1) + \widehat{u}.\widehat{v} + \widehat{u} \cdot (\widehat{u} \times \widehat{v}) }{\overrightarrow{A} \cdot \widehat{u} = \lambda + \frac{1}{2}}$$

$$\begin{array}{r} \Rightarrow 2\overrightarrow{\text{ }A} \cdot \widehat{u} = 2\lambda + 1\#(2) \end{array}$$

Dot equation (1) with $\widehat{v}$

$${\overrightarrow{A} \cdot \widehat{v} = \lambda(\widehat{u} \cdot \widehat{v}) + \widehat{v} \cdot \widehat{v} + \widehat{v} \cdot (\widehat{u} \times \widehat{v}) }{\overrightarrow{A} \cdot \widehat{v} = \frac{\lambda}{2} + 1}$$

$$\begin{array}{r} \overrightarrow{A} \cdot \widehat{v} - \frac{\lambda}{2} = 1\#(3) \end{array}$$

From (2) and (3)

$${2\overrightarrow{\text{ }A} \cdot \widehat{u} - 2\lambda = \overrightarrow{A} \cdot \widehat{v} - \frac{\lambda}{2} }{\therefore\lambda = \frac{4}{3}\overrightarrow{\text{ }A} \cdot \widehat{u} - \frac{2}{3}\overrightarrow{\text{ }A} \cdot \widehat{v} }$$