Determine if the following pair of planes is parallel, orthogonal, or neither parallel nor orthogonal. \[ 3 x+2 y-3 z=29 \text { and }-6 x-10 y+z=29 \] Choose the correct answer below. A. The planes are parallel. B. The planes are orthogonal. C. The planes are neither parallel nor orthogonal.

To determine the relationship between the two planes given by their equations, we'll first examine the normal vectors of each plane. The first plane \( 3x + 2y - 3z = 29 \) has a normal vector \( \mathbf{n_1} = \langle 3, 2, -3 \rangle \). The second plane \( -6x - 10y + z = 29 \) has a normal vector \( \mathbf{n_2} = \langle -6, -10, 1 \rangle \). Now, we check if the planes are parallel by comparing the direction of their normal vectors. If the normal vectors are scalar multiples of each other, the planes are parallel. Calculating the dot product \( \mathbf{n_1} \cdot \mathbf{n_2} \) will help determine if they are orthogonal. If the dot product is zero, the planes are orthogonal. Calculating the dot product: \[ 3(-6) + 2(-10) + (-3)(1) = -18 - 20 - 3 = -41 \] Since the dot product is not zero, the planes are not orthogonal. Thus, the correct answer is C. The planes are neither parallel nor orthogonal.