Question

Let $\overrightarrow{a}$ be a unit vector and $\overrightarrow{b}$ be a non-zero vector not parallel to $\overrightarrow{a}$. If two sides of a triangle are represented by the vectors $\sqrt{3}(\overrightarrow{a}\times \overrightarrow{b})$ and $\overrightarrow{b}-(\overrightarrow{a}\cdot \overrightarrow{b})\overrightarrow{a}$, then the angles of the triangle are

• A. ${90}^{\circ },{60}^{\circ },{30}^{\circ }$
• B. ${45}^{\circ },{45}^{\circ },{90}^{\circ }$
• C. ${60}^{\circ },{60}^{\circ },{60}^{\circ }$
• D. ${75}^{\circ },{45}^{\circ },{60}^{\circ }$

Answer 1

The correct option is A ${90}^{\circ },{60}^{\circ },{30}^{\circ }$

Side 1: $\sqrt{3}(\overrightarrow{a}\times \overrightarrow{b})$
Side 2: $\overrightarrow{b}-(\overrightarrow{a}\cdot \overrightarrow{b})\overrightarrow{a}=(\overrightarrow{a}\cdot \overrightarrow{a})\overrightarrow{b}-(\overrightarrow{a}\cdot \overrightarrow{b})\overrightarrow{a}$
Hence, Side 2: $(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{a}$
Therefore, two sides are $\sqrt{3}(\overrightarrow{a}\times \overrightarrow{b}),\left(\right(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{a})$
We observe that $\sqrt{3}(\overrightarrow{a}\times \overrightarrow{b})\cdot \left(\right(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{a})=0$
$\therefore$ Angle between these two sides is ${90}^{\circ }.$
Length of these two sides are in the ratio $=\frac{|\sqrt{3}(\overrightarrow{a}\times \overrightarrow{b})|}{|(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{a}|}=\frac{\sqrt{3}|\overrightarrow{a}\times \overrightarrow{b}|}{|\overrightarrow{a}\times \overrightarrow{b}||\overrightarrow{a}||\sin {90}^0|}=\frac{\sqrt{3}}{1}=\tan \theta$ (where $\theta ,({90}^{\circ }-\theta )$ represent the angles between sides $1,3$ and $2,3$)
$\therefore$ The remaining angles are $\theta ={60}^{\circ },({90}^{\circ }-\theta )={30}^{\circ }.$