Final answer:
To find the required unit vector, spherical to Cartesian conversion formulas are used with the given angles π/3 and π/6, leading to a unique unit vector with Cartesian components (1/4, √3/4, √3/2).
Step-by-step explanation:
To find all unit vectors that make an angle of π/3 with the positive part of the x-axis and π/6 with the positive part of the z-axis, we use spherical coordinates. A unit vector in a three-dimensional space can be described by two angles and its length. Since we are looking for unit vectors, the length is 1. The angles in spherical coordinates are typically the angle θ from the positive z-axis and the angle φ in the xy-plane from the positive x-axis.
In this problem, the angle from the z-axis (θ) is π/6 and the angle from the x-axis (φ) is π/3. Using these angles, we can calculate the Cartesian components of the unit vector â using the spherical to Cartesian conversion formulas:
- x = cos(φ) * sin(θ)
- y = sin(φ) * sin(θ)
- z = cos(θ)
When φ = π/3 and θ = π/6, we get:
- x = cos(π/3) * sin(π/6) = (1/2) * (1/2) = 1/4
- y = sin(π/3) * sin(π/6) = (√3/2) * (1/2) = √3/4
- z = cos(π/6) = √3/2
Hence, the unit vector with the specified angles is:
â = (1/4)î + (√3/4)ê + (√3/2)ê
Remember, this vector represents a single unique unit vector that satisfies the given angle constraints.