Final answer:
To convert (1,0,−3) to spherical coordinates, calculate the radial distance ρ as the square root of the sum of the squares of x, y, and z; θ as the arctangent of y over x, which is 0 for this point; and φ as the arccosine of z over ρ.
Step-by-step explanation:
To convert the point (x,y,z)=(1,0,−3) to spherical coordinates, we use the relationships between rectangular and spherical coordinates. In spherical coordinates, the radial distance ρ is the distance from the point to the origin, θ is the azimuthal angle measured from the positive x-axis towards the y-axis in the xy-plane, and φ is the polar angle measured from the positive z-axis.
The radial distance ρ is calculated using the formula ρ = √(x2 + y2 + z2).
θ is the azimuthal angle, which can be calculated using the formula θ = atan2(y,x).
φ is the polar angle, which is calculated using the formula φ = acos(z/ρ).
For the point (1,0,−3), ρ = √(12 + 02 + (−3)2) = √(10), θ = atan2(0,1) = 0 (since the point lies on the x-axis), and φ = acos(−3/√(10)). We can leave φ in terms of acos or evaluate it to a decimal approximation.