Question

If a and b are two unit vectors the what is the angle between a and b so that √3a-b be a unit vector

Answer 1

$\textbf{Given:}$

$\text{ \overrightarrow{a} , \overrightarrow{b} and \sqrt{3}\overrightarrow{a}-\overrightarrow{b} are unit vectors}$

$\text{Then, |\overrightarrow{a}| = |\overrightarrow{b}| = |\sqrt{3}\overrightarrow{a}-\overrightarrow{b}| =1}$

$\implies|\sqrt{3}\overrightarrow{a}-\overrightarrow{b}|^2=1^2}$

$\text{We know that,}$

$\boxed{\bf|\overrightarrow{a}-\overrightarrow{b}|^2=|\overrightarrow{a}|^2+|\overrightarrow{b}|^2-2\,|\overrightarrow{a}|\,|\overrightarrow{b}|\,cos\theta}$

$\implies\,3|\overrightarrow{a}|^2+|\overrightarrow{b}|^2-2\sqrt{3}\,|\overrightarrow{a}|\,|\overrightarrow{b}|\,cos\theta=1$

$\implies\,3(1)^2+(1)^2-2\sqrt{3}(1)(1)cos\theta=1$

$\implies\,4-2\sqrt{3}cos\theta=1$

$\implies\,2\sqrt{3}cos\theta=3$

$\implies\,2cos\theta=\sqrt{3}$

$\implies\,cos\theta=\frac{\sqrt{3}}{2}$

$\implies\boxed{\bf\theta=\frac{\pi}{6}}$

$\therefore\textbf{The angle between \overrightarrow{a} and \overrightarrow{b} is \frac{\pi}{6} }$

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If |a| = 13,|b|= 5 and a.b = 60, then

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