Final answer:
The third direction angle of the vector, given that the first two angles are π/2 and π/4 and the third angle must be acute, is found to be π/4 or 45°.
Step-by-step explanation:
The student has provided two direction angles of a vector and needs to determine the third direction angle, while ensuring that the angle α is acute (less than 90°). The known direction angles are π/2 and π/4. In three-dimensional space, a vector's direction angles α, β, and γ with the x, y, and z axes respectively must satisfy the following relationship:
cos²(α) + cos²(β) + cos²(γ) = 1
Given the first direction angle is π/2, which has a cosine of 0, and the second is π/4, which has a cosine of √2/2, we can find the cosine of the third angle γ:
cos²(γ) = 1 - cos²(π/4)
cos²(γ) = 1 - (√2/2)²
cos²(γ) = 1 - 1/2
cos²(γ) = 1/2
Hence, cos(γ) = √(1/2) or -√(1/2). Since we need γ to be either obtuse or acute, and α is acute, we conclude γ must also be acute (we cannot have all angles acute or obtuse), we select the positive value:
cos(γ) = √(1/2)
Therefore, γ = π/4 or 45°.
Finally, we can say that the third direction angle γ is 45° or π/4 radians, and it is acute.