Question

P,q,r,s are vector of equal magnitude. if p+q-r=0 angle between p and q is tita1.if p+q-s=0 angle between p and s is tita2. the ratio of tita1 and tita2

Answer 1

Answer:

2

Explanation:

Given

p, q, r, and s are vectors of equal magnitude.

Let |p| = |q| = |r| = |s| = a units

Given

p + q - r = 0

⇒p +q = r

Squaring on both sides

⇒|p + q|² = |r|²

⇒|p|² + |q|² + 2 p.q = |r|²   [∵|a+b|² = |a|²+|b|²+2a.b where a.b is dot product

                                            of vectors a and b]

⇒a² + a² + 2|p||q| cos(p,q) = a²   [∵|p| = |q| = |r| = |s| = a units]

⇒2a² + 2a² cos θ₁ = a²             [∵ Angle between p and q is θ₁]

⇒ 2 + 2 cos θ₁ = 1

⇒ 2 cos θ₁ = -1

⇒cos θ₁ = -1/2

⇒cos θ₁ = cos 2π/3

∴ θ₁ = 2π/3  ___(1)

Given

p + q - s = 0

⇒p - s = -q

Squaring on both sides

⇒(p - s)² = (-q)²

⇒|p|² + |s|² - 2p.s = |q|²  [∵ |a-b|²=|a|²+|b|²-2a.b where a.b is dot product]

⇒a² + a² - 2|p||s| cos (p,s) = a²    [∵|p| = |q| = |r| = |s| = a units]

⇒2a² - 2a² cos θ₂ = a²               [∵ Angle between p and s is θ₂]

⇒2 - 2 cos θ₂ = 1

⇒2 cos θ₂ = 2 - 1

⇒2 cos θ₂ = 1

⇒cos θ₂ = 1/2

⇒cos θ₂ = cos π/3

∴ θ₂ = π/3  ___(2)

∴θ₁: θ₂ = θ₁/ θ₂ = (2π/3)/(π/3)

∴θ₁: θ₂ = 2