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

To convert spherical coordinates ( B(r, theta, phi) = B(4, 25^circ, 120^circ) ) into Cartesian coordinates ( (x, y, z) ), we use the following formulas:

1. ( x = r cdot sin(theta) cdot cos(phi) )
2. ( y = r cdot sin(theta) cdot sin(phi) )
3. ( z = r cdot cos(theta) )

Where:

– ( r ) is the radius,

– ( theta ) is the polar angle (measured from the positive z-axis),

– ( phi ) is the azimuthal angle (measured from the positive x-axis in the x-y plane).

Given:

– ( r = 4 )

– ( theta = 25^circ )

– ( phi = 120^circ )

First, convert angles from degrees to radians:

– ( theta = 25^circ = frac{25 pi}{180} approx 0.436 , text{radians} )

– ( phi = 120^circ = frac{120 pi}{180} = frac{2pi}{3} approx 2.094 , text{radians} )

Now, calculate the Cartesian coordinates:

1.

To convert the Cartesian coordinates (x, y, z) = (3, 4, 5) to spherical coordinates (ρ, θ, φ), we use the following formulas:

1. ( ρ = sqrt{x^2 + y^2 + z^2} )
2. ( θ = tan^{-1}left(frac{y}{x}right) )
3. ( φ = cos^{-1}left(frac{z}{ρ}right) )

Now plug in the values:

1. Calculate ( ρ ):

[

ρ = sqrt{3^2 + 4^2 + 5^2} = sqrt{9 + 16 + 25} = sqrt{50} = 5sqrt{2}

]

2. Calculate ( θ ):

[

θ = tan^{-1}left(frac{4}{3}right) approx 0.927 text{ radians} quad (text{or } 53.13^circ)

]

3. Calculate ( φ ):

[

φ = cos^{-1}left(frac{5}{5sqrt{2}}right) = cos^{-1}left(frac{1}{sqrt{2}}right) = frac{pi}{4} text{ radians} quad (text{or } 45