Transform the vector (4,-2,-4) at (1,2,3) into spherical coordinates.

To transform the vector (4, -2, -4) at the point (1, 2, 3) into spherical coordinates, we first need to find the coordinates in relation to the origin. The spherical coordinates (r, θ, φ) are given by:

1. r (the radial distance) = √(x² + y² + z²)
2. θ (the azimuthal angle) = atan2(y, x)
3. φ (the polar angle) = acos(z/r)

First, we adjust the vector (4, -2, -4) based on the point (1, 2, 3):

– The vector is effectively at the point (1+4, 2-2, 3-4) = (5, 0, -1).

Now, we calculate the spherical coordinates:

1. Calculate r:

r = √(5² + 0² + (-1)²) = √(25 + 0 + 1) = √26.

2. Calculate θ:

θ = atan2(0, 5) = 0 (since y=0 and x>0).

3. Calculate φ:

φ = acos(-1/√26).

So, the spherical coordinates are:

– ( r = sqrt{26} )

– ( θ = 0 )

– ( φ = acos