Question

In Exercises \(25-28,\) convert the point from spherical coordinates to rectangular coordinates. $$ (12,-\pi / 4,0) $$

Short answer

The rectangular coordinates of the point is (0, 0, 12).

Step-by-step solution

Understand Spherical Coordinates

A point in three dimensions can be represented in spherical coordinates as \( (r, \theta, \phi) \), where \( r \) is the distance from the origin to the point, \( \theta \) is the angle measured from the positive x-axis (in the x-y plane), and \( \phi \) is the angle between the positive z-axis and the line segment from the origin to the point.

Understand Conversion Formulas

In the given spherical coordinates, \(r = 12\), \(\theta = - \frac{\pi}{4}\), and \(\phi = 0\). The formulas for converting spherical to rectangular coordinates are:\[x = r \sin\phi \cos\theta\]\[y = r \sin\phi \sin\theta\]\[z = r \cos\phi\]

Apply Conversion Formulas

Substitute the given spherical coordinates in the conversion formulas:\[x = 12 \sin(0) \cos(-\frac{\pi}{4}) = 12 * 0 * \frac{\sqrt{2}}{2} = 0\]\[y = 12 \sin(0) \sin(-\frac{\pi}{4}) = 12 * 0 * -\frac{\sqrt{2}}{2} = 0\]\[z = 12 \cos(0) = 12 * 1 = 12\]