Question

If$\overrightarrow{P} + \overrightarrow{Q} + \overrightarrow{R} = 0$ and out of these, two vectors are equal in magnitude and the third vector has magnitude √2 times that of 2any of these two vectors, then angles among the three vectors are

• A. 45⁰, 75⁰, 75⁰
• B. 45⁰, 90⁰, 135⁰
• C. 90⁰, 135⁰, 180⁰
• D. 90⁰, 135⁰, 135⁰

Answer 1

Answer: (D)

90⁰, 135⁰, 135⁰

Answer 2

Let P = Q = x and R = √2x Using $\overrightarrow{P} + \overrightarrow{Q} + \overrightarrow{R} = 0$ we get, $\overrightarrow{P} + \overrightarrow{Q} = - \overrightarrow{C}$

($\overrightarrow{P} + \overrightarrow{Q}$). ($\overrightarrow{P} + \overrightarrow{Q}$) = ($\overrightarrow{R}$) .(-$\overrightarrow{R}$) Then P² + Q² + 2PQ cos θ = R² i.e., x² + x² + 2x² cos θ = 2x² i.e., cosθ = 90⁰ Again $\overrightarrow{Q} + \overrightarrow{R}$ = - $\overrightarrow{P}$

($\overrightarrow{Q} + \overrightarrow{R}$). ($\overrightarrow{Q} + \overrightarrow{R}$) = ($\overrightarrow{P}$). (-$\overrightarrow{P}$) Then Q² + R² + 2QR

cos α = P² i.e, cos α = -1/2 √2 or φ = 135⁰ Third angle = 360 – (135 + 90) = 360 – 225 = 135⁰