Question

Let a, b, c ∈ R be such that a² + b² + c² = 1. If `a cos θ = b cos(θ + (2π)/3) = c cos(θ + (4π)/3)`, where θ = `π/9`, then the angle between the vectors `ahati + bhatj + chatk` and `bhati + chatj + ahatk` is ______.

Options

• `π/2`

• `(2π)/3`

• `π/9`

• 0

Answer 1

Let a, b, c ∈ R be such that a² + b² + c² = 1. If `a cos θ = b cos(θ + (2π)/3) = c cos(θ + (4π)/3)`, where θ = `π/9`, then the angle between the vectors `ahati + bhatj + chatk` and `bhati + chatj + ahatk` is `underlinebb(π/2)`.

Explanation:

Let a cos θ = `b cos(θ + (2π)/3) = c cos(θ + (4π)/3)` = k

then, a = `k/cosθ`, b = `k/(cos(θ + (2π)/3)`, c = `k/(cos(θ + (4π)/3)`

Given a² + b² + c² = 1

Let `vecp = ahati + bhatj + chatk`

`vecq = bhati + chatj + ahatk`

now

`vecp.vecq = |vecp||vecq|cosα`

`\implies` ab + bc + ca = `sqrt(a^2 + b^2 + c^2). sqrt(b^2 + c^2 + a^2) cos α`

`\implies` cos α = ab + bc + ca

ab + bc + ca = `k^2([cos(θ + (4π)/3) + cos θ + cos(θ + (2π)/3)])/(cos(θ + (4π)/3).cosθ.cos(θ + (2π)/3)`

= `k^2[(cosθ + 2cos(θ + π). cos(π/3))/(cosθ.cos(θ + (2π)/3).cos(θ + (4π)/3))]` 

= `k^2[(cosθ - 2cosθ. 1/2)/(cosθ.cos(θ + (2π)/3).cos(θ + (4π)/3))]` 

cos α = `((ahati + bhatj + chatk).(bhati + chatj + ahatk))/(sqrt(a^2 + b^2 + c^2).sqrt(b^2 + c^2 + a^2))`

= ab + bc + ca 

= 0

α = `π/2`