Given vector R = 3 ax - 4 ay - 2az at point P(4,5 , 2)Determine Rɸ
Question
Given vector R = 3 ax - 4 ay - 2az at point P(4,5 , 2)Determine Rɸ
Solution
To determine Rɸ, we first need to convert the Cartesian coordinates to spherical coordinates.
The spherical coordinates (r, θ, φ) are related to the Cartesian coordinates (x, y, z) by the following equations:
r = sqrt(x² + y² + z²) θ = arccos(z/r) φ = arctan(y/x)
Given the point P(4,5,2), we can calculate r, θ, and φ as follows:
r = sqrt(4² + 5² + 2²) = sqrt(16 + 25 + 4) = sqrt(45) θ = arccos(2/sqrt(45)) = arccos(2/6.71) = 1.23 radians φ = arctan(5/4) = 0.896 radians
Now, we can express the vector R in spherical coordinates. The unit vectors in spherical coordinates are ar, aθ, and aφ.
R = 3ax - 4ay - 2az can be written as R = 3ar - 4aθ - 2aφ.
Finally, Rφ is the φ-component of the vector R, which is -2.
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