The sum of two forces $\vec{P}$ and $\vec{Q}$ is $\vec{R}$ such that $|\vec{R}|=|\vec{P}|$. The angle $\theta$ (in degrees) that the resultant of $2 \vec{P}$ and $\vec{Q}$ will make with $\vec{Q}$ is, ______.

JEE Main · 2020 · 07 Jan 2020 · Shift 2 · Vector

The sum of two forces

P

and

Q

R

such that

∣ R ∣ = ∣ P ∣

. The angle

θ

(in degrees) that the resultant of

2 P

and

Q

will make with

Q

is, ______.

Answer: 90

Squaring

∣ P + Q ∣ = ∣ P ∣

gives

P^{2} + Q^{2} + 2 P Q cos θ = P^{2}

, so

Q + 2 P cos θ = 0

. Now the resultant of

2 P

and

Q

makes an angle

α

with

Q

where

tan α = \frac{2 P s i n θ}{Q + 2 P c o s θ}

. The denominator is exactly the quantity just shown to vanish, so

tan α \to \infty

and

α = 9 0^{\circ}

JEE Main · 2020 · 07 Jan 2020 · Shift 2 · Integer answer