Question

If a vector has direction angles α = π/4 and β = π/3, find the third direction angle γ

Answer 1

The third direction angle γ of a vector with direction angles α = π/4 and β = π/3 is either π/3 or 4π/3, depending on the vector's quadrant.

Step-by-step explanation:

If a vector has direction angles α = π/4 and β = π/3, to find the third direction angle γ, we can use the fact that these angles are related to the vector components along the x, y, and z axes. In a three-dimensional Cartesian coordinate system, any vector's direction angles with respect to the positive x, y, and z axes satisfy the equation:

cos^2(α) + cos^2(β) + cos^2(γ) = 1

We already have α = π/4 and β = π/3. Plugging these values into the equation gives:

• cos^2(π/4) = (1/√2)^2 = 1/2
• cos^2(π/3) = (1/2)^2 = 1/4

Now, let's find cos^2(γ):

cos^2(γ) = 1 - (1/2) - (1/4)

cos^2(γ) = 1/4 = (1/2)^2

So, γ = cos^{-1}(1/2) or γ = cos^{-1}(-1/2), since cosine can be positive or negative depending on the quadrant where the vector is located. Therefore,

π/3 ≤ γ ≤ 2π/3 for γ in the first or the fourth quadrant giving us γ = π/3, and,

π + π/3 ≤ γ ≤ π + 2π/3 for γ in the second or third quadrant giving us γ = 4π/3.

Therefore, the third direction angle γ is either π/3 or 4π/3 depending on the specific quadrant in which the vector resides.

Answer 2

For a vector:
cos^2(α) + cos^2(β) + cos^2(γ) = 1

From the givens:
α = π/4, therefore, cos(α) = 1/root 2 and cos^2(α) = 1/2
β = π/3, therefore, cos(β) = 1/2 and cos^2(β) = 1/4

Substitute in the above equation:
1/2 + 1/4 + cos^2(γ) = 1
cos^2(γ) = 1 - 1/2 - 1/4
cos^2(γ) = 1/4
cos(γ) is either 1/2 or -1/2

For cos(γ)= 1/2 .......> γ = π/3
For cos(γ)= -1/2 .......> γ = 2π/3