Question

If a vector r=x i cap +y j cap +z k cap, makes angle pi /3,pi/3and pi/n with x-axis, y-axis and z-axis respectively. Find the value of n.

Answer 1

vector a = x i + y j + z k makes angle π/3 , π/4 and π/n with x - axis , y - axis and z - axis respectively.

To find : The value of n.

solution : we know, when any vector a makes angles α, β and γwith x - axis , y - axis and z - axis respectively then cosα, cosβ and cosγ are known as direction ratios.

we know, cos²α + cos²β + cos²γ = 1

here α = π/3 ⇒cosα = cosπ/3 = 1/2

β = π/4 ⇒cosβ = cos(π/4) = 1/√2

and γ = π/n ⇒cosγ = cos(π/n)

now (1/2)² + (1/√2)² + cos²(π/n) = 1

⇒3/4 + cos²(π/n) = 1

⇒cos²(π/n) = 1 - 3/4 = 1/4

⇒cos²(π/n) = (1/2)² = cos²(π/3)

⇒π/n = π/3

so n = 3

Therefore the value of n = 3

Answer 2

Answer:

n=3

Explanation:

cos²α + cos²β + cos²γ = 1

α = π/3 ⇒cosα = cosπ/3 = 1/2

β = π/4 ⇒cosβ = cos(π/4) = 1/√2

γ = π/n ⇒cosγ = cos(π/n)

(1/2)² + (1/√2)² + cos²(π/n) = 1

⇒3/4 + cos²(π/n) = 1

⇒cos²(π/n) = 1 - 3/4 = 1/4

⇒cos²(π/n) = (1/2)² = cos²(π/3)

⇒π/n = π/3

n = 3