Final answer:
To convert from rectangular to spherical coordinates, use the formulas x = r sin(θ) cos(φ), y = r sin(θ) sin(φ), and z = r cos(θ). Given the rectangular coordinates (-5√2, 5√2, 10√3), substitute the values into the formulas to obtain the spherical coordinates (20, π/6, −π/4).
Step-by-step explanation:
To convert from rectangular to spherical coordinates, we can use the formulas:
x = r sin(θ) cos(φ)
y = r sin(θ) sin(φ)
z = r cos(θ)
Given the rectangular coordinates (-5√2, 5√2, 10√3), we can substitute the values into the formulas to obtain the spherical coordinates.
r = √(x² + y² + z²) = √((-5√2)² + (5√2)² + (10√3)²) = √(50 + 50 + 300) = √400 = 20
θ = arccos(z/r) = arccos(10√3/20) = arccos(√3/2) = π/6
φ = arctan(y/x) = arctan((5√2)/(−5√2)) = arctan(-1) = −π/4
Therefore, the spherical coordinates for the given rectangular coordinates are (20, π/6, −π/4).