Question

Assertion :If $\hat{u}$ and $\hat{v}$ are unit vectors inclined at an angle $\alpha$ and $\hat{a}$ is a unit vector bisecting the angle between them, then $\hat{a}=\frac{\hat{u}+\hat{v}}{2\cos \left(\alpha /2\right)}$ Reason: If $ABC$ is an isosceles triangle with $AB=AC=1$, then the vector representing bisector of angle $A$ is $\frac{\overrightarrow{AB}+\overrightarrow{AC}}{2}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

In an isosceles triangle $ABC$ in which $AB=AC,$ the median and bisector from $A$ must be same line so statement $2$ is true
$AD=\frac{u+v}{2}$
${\left|AD\right|}^2=.\frac{1}{4}[{\left|u\right|}^2+{\left|v\right|}^2+2uv]$
$=\frac{1}{4}[2+2\cos \alpha ]$
$={\cos }^2\alpha /2$
$\Rightarrow$ Unit vector along $AD=\frac{1}{2\cos \left(\alpha /2\right)}(u+v)$