Determine if the following planes are parallel, orthogonal, or neither parallel nor orthogonal. \( 4 x+4 y+2 z=3 \) and \( -2 x-3 y+10 z=3 \) The planes are orthogonal. The planes are neither parallel nor orthogonal. The planes are parallel.

To determine the relationship between the two planes, we first need to find their normal vectors from the equations given. For the first plane \( 4x + 4y + 2z = 3 \), the normal vector is \( \vec{n_1} = (4, 4, 2) \). For the second plane \( -2x - 3y + 10z = 3 \), the normal vector is \( \vec{n_2} = (-2, -3, 10) \). Next, we check if the planes are parallel by examining the ratios of their normal vectors. If the normal vectors are scalar multiples of each other, the planes are parallel. The planes are not parallel as there is no constant \( k \) making \( \vec{n_1} = k \vec{n_2} \). To check for orthogonality, we find the dot product of the two normal vectors: \[ \vec{n_1} \cdot \vec{n_2} = 4 \cdot (-2) + 4 \cdot (-3) + 2 \cdot 10 = -8 - 12 + 20 = 0. \] Since the dot product is zero, the planes are orthogonal. Thus, the correct relationship between the planes is that they are orthogonal.