Question

At what angle the two vectors of magnitude (P+Q) and (P-Q) should act so that the resultant is √P^2+Q^2?

Answer 1

Given :-

• Two vectors of magnitudes (P+Q) and (P-Q) have a resultant of √(P²+Q²)

To find :-

• Angle formed between the two vectors

Formula used :-

• Formula to calculate the Resultant of two vectors

$\blue{\bigstar}\boxed{\sf{R^2=M^2+N^2+2\;MN\;cos\;\theta}}$

(where R is the magnitude of resultant of two vectors M and N and θ is the angle formed by vectors M and N)

Solution :-

Using the formula for calculating Resultant of two vectors

↦ [√(P²+Q²)]² = (P+Q)² + (P-Q)² + 2 (P+Q)(P-Q) cos θ

↦ (P²+Q²) = (P+Q)² + (P-Q)² + 2 (P+Q)(P-Q) cos θ

↦ P² + Q² = P² + Q² + 2 PQ + P² + Q² - 2 PQ + 2 (P² - Q²) cos θ

↦ P² + Q² = 2 (P² + Q²) + 2 (P² - Q²) cos θ

↦ 2 (P² - Q²) cos θ = P² + Q² - 2 (P² + Q²)

↦ 2 ( P²- Q²) cos θ = - (P² + Q²)

$\mapsto\sf{cos\;\theta=\dfrac{-(P^2+Q^2)}{2(P^2-Q^2)}}$

$\longmapsto\boxed{\boxed{\sf{\theta=cos^{-1}\left(\dfrac{P^2+Q^2}{2(Q^2-P^2)}\right)}}}$