Question

Given that $\overrightarrow{P}+\overrightarrow{Q}+\overrightarrow{R}=\overrightarrow{0}$. Two out of the three vectors are equal in magnitude of the third vector is $\sqrt{2}$ times that of the other two. Which of the following can be the angles between these vectors?

• A. ${90}^o,{135}^o,{135}^o$
• B. ${45}^o,{45}^o,{90}^o$
• C. ${30}^o,{60}^o,{90}^o$
• D. ${45}^o,{90}^o,{135}^o$

Answer 1

The correct option is A ${90}^o,{135}^o,{135}^o$

Sum of the vectors is zero, which implies that they form a triangle.

Given that, $r=\sqrt{2\left(p\right)}$

$⟹p^2+q^2=2\times \left(p^2\right)=r^2$

Hence, the triangle formed by $\overrightarrow{P},$ $\overrightarrow{Q}$ and $\overrightarrow{R}$ obey the Pythagoras theorem. Hence it is a right-angled triangle with $\overrightarrow{R}$ as the hypotenuse.

Therefore, the angles between the vectors are ${90}^{\circ },{135}^{\circ }$ and ${135}^{\circ }$.