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To convert the vector ((4, -2, -4)) at the point ((1, 2, 3)) into spherical coordinates, we first need to find the origin of the vector in Cartesian coordinates:
1. Start point ((1, 2, 3)).
2. Direction vector ((4, -2, -4)).
3. End point can be found by adding the direction vector to the start point:
((1+4, 2-2, 3-4) = (5, 0, -1)).
Now, we convert the point ((5, 0, -1)) to spherical coordinates ((r, theta, phi)):
1. Calculate (r), the distance from the origin:
[
r = sqrt{x^2 + y^2 + z^2} = sqrt{5^2 + 0^2 + (-1)^2} = sqrt{25 + 0 + 1} = sqrt{26}
]
2. Calculate (theta), the azimuthal angle, which is the angle in the xy-plane from the positive x-axis:
[
theta = tan^{-1}left(frac{y}{x}right) = tan^{-1}left(frac{0}{5}right) = 0
]
3
Answer: b
Explanation: r = √(x2+y2+z2) = 3.74
Θ = cos-1(z/r) = cos-1(3/3.74) = 36.7⁰
Φ = tan-1(y/x) = tan-1(2/1) = 63.4⁰
A = (4 sin θ cos φ – 2 sin θ sin φ – 4cos θ)i + (4 cos θ cos φ – 2 cos θ sin φ + 4 sinθ)j (-4 sin φ – 2 cos φ)k
On substituting r, θ, φ, A = -3.197i + 2.393j – 4.472k.