Convert the point \( (\rho, \theta, \phi)=\left(6, \frac{3 \pi}{4}, \pi\right) \) to Cartesian coordinates. Give answers either as expressions, or decimals to at least one decimal place. \( (x, y, z)=\square \) Question Help: \( \square \) Video 1

The conversion formulas from spherical to Cartesian coordinates are \[ x = \rho \sin(\phi) \cos(\theta), \quad y = \rho \sin(\phi) \sin(\theta), \quad z = \rho \cos(\phi). \] Given \(\rho = 6\), \(\theta = \frac{3\pi}{4}\), and \(\phi = \pi\): 1. Calculate \(x\): \[ x = 6 \sin(\pi) \cos\left(\frac{3\pi}{4}\right). \] Since \(\sin(\pi) = 0\), \[ x = 6 \cdot 0 \cdot \cos\left(\frac{3\pi}{4}\right) = 0. \] 2. Calculate \(y\): \[ y = 6 \sin(\pi) \sin\left(\frac{3\pi}{4}\right). \] Again, \(\sin(\pi) = 0\), \[ y = 6 \cdot 0 \cdot \sin\left(\frac{3\pi}{4}\right) = 0. \] 3. Calculate \(z\): \[ z = 6 \cos(\pi). \] Since \(\cos(\pi) = -1\), \[ z = 6 \cdot (-1) = -6. \] Thus, the Cartesian coordinates are \[ (x, y, z) = (0, 0, -6). \]