In the following determine whether the given planes are parallel or perpendicular and in case they are neither, find the angles between them :
(i) 7x + 5y + 6z + 30 = 0 and 3x - y - 10z + 4 = 0
(ii) 2x + y + 3z - 2 = 0 and x - 2y + 5 = 0
(iii) 2x - 2y + 4z + 5 = 0 and 3x - 3y + 6z - 1 = 0
(iv) 2x - y + 3z - 1 = 0 and 2x - y + 3z + 3 = 0
(v) 4x + 8y + z - 8 = 0 and y + z - 4 = 0.

Answer

To determine whether the given planes are parallel or perpendicular, and to find the angles between them if they are neither, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Normal Vectors**:
For each plane given in the form \( ax + by + cz + d = 0 \), the normal vector can be identified as \( \vec{n} = (a, b, c) \).

2. **Calculate the Dot Product**:
For two planes with normal vectors \( \vec{n_1} = (a_1, b_1, c_1) \) and \( \vec{n_2} = (a_2, b_2, c_2) \), the dot product is given by:
\[ \vec{n_1} \cdot \vec{n_2} = a_1a_2 + b_1b_2 + c_1c_2 \]

3. **Check for Perpendicularity**:
If the dot product is zero, the planes are perpendicular.

4. **Check for Parallelism**:
The planes are parallel if the ratios of the coefficients of \( x, y, z \) are equal:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

5. **Calculate the Angle**:
If the planes are neither parallel nor perpendicular, the angle \( \theta \) between the planes can be found using:
\[ \cos \theta = \frac{\vec{n_1} \cdot \vec{n_2}}{|\vec{n_1}| |\vec{n_2}|} \]
where \( |\vec{n_1}| \) and \( |\vec{n_2}| \) are the magnitudes of the normal vectors.

Now, let's apply these steps to each of the given pairs of planes.

### (i) Planes: \( 7x + 5y + 6z + 30 = 0 \) and \( 3x - y - 10z + 4 = 0 \)

- **Normal Vectors**:
\( \vec{n_1} = (7, 5, 6) \) and \( \vec{n_2} = (3, -1, -10) \)

- **Dot Product**:
\[ \vec{n_1} \cdot \vec{n_2} = 7 \cdot 3 + 5 \cdot (-1) + 6 \cdot (-10) = 21 - 5 - 60 = -44 \]

- **Check for Perpendicularity**:
Since \(-44 \neq 0\), the planes are not perpendicular.

- **Check for Parallelism**:
\[ \frac{7}{3}, \frac{5}{-1}, \frac{6}{-10} \quad \text{(not equal)} \]
Hence, the planes are not parallel.

- **Calculate the Angle**:
\[ |\vec{n_1}| = \sqrt{7^2 + 5^2 + 6^2} = \sqrt{49 + 25 + 36} = \sqrt{110} \]
\[ |\vec{n_2}| = \sqrt{3^2 + (-1)^2 + (-10)^2} = \sqrt{9 + 1 + 100} = \sqrt{110} \]
\[ \cos \theta = \frac{-44}{\sqrt{110} \cdot \sqrt{110}} = \frac{-44}{110} = -\frac{2}{5} \]
\[ \theta = \cos^{-1}\left(-\frac{2}{5}\right) \]

### (ii) Planes: \( 2x + y + 3z - 2 = 0 \) and \( x - 2y + 5 = 0 \)

- **Normal Vectors**:
\( \vec{n_1} = (2, 1, 3) \) and \( \vec{n_2} = (1, -2, 0) \)

- **Dot Product**:
\[ \vec{n_1} \cdot \vec{n_2} = 2 \cdot 1 + 1 \cdot (-2) + 3 \cdot 0 = 2 - 2 + 0 = 0 \]
The planes are perpendicular.

### (iii) Planes: \( 2x - 2y + 4z + 5 = 0 \) and \( 3x - 3y + 6z - 1 = 0 \)

- **Normal Vectors**:
\( \vec{n_1} = (2, -2, 4) \) and \( \vec{n_2} = (3, -3, 6) \)

- **Dot Product**:
\[ \vec{n_1} \cdot \vec{n_2} = 2 \cdot 3 + (-2) \cdot (-3) + 4 \cdot 6 = 6 + 6 + 24 = 36 \]

- **Check for Parallelism**:
\[ \frac{2}{3} = \frac{-2}{-3} = \frac{4}{6} \quad \text{(equal)} \]
Hence, the planes are parallel.

### (iv) Planes: \( 2x - y + 3z - 1 = 0 \) and \( 2x - y + 3z + 3 = 0 \)

- **Normal Vectors**:
\( \vec{n_1} = (2, -1, 3) \) and \( \vec{n_2} = (2, -1, 3) \)

- **Check for Parallelism**:
Since the normal vectors are identical, the planes are parallel.

### (v) Planes: \( 4x + 8y + z - 8 = 0 \) and \( y + z - 4 = 0 \)

- **Normal Vectors**:
\( \vec{n_1} = (4, 8, 1) \) and \( \vec{n_2} = (0, 1, 1) \)

- **Dot Product**:
\[ \vec{n_1} \cdot \vec{n_2} = 4 \cdot 0 + 8 \cdot 1 + 1 \cdot 1 = 0 + 8 + 1 = 9 \]

- **Check for Parallelism**:
Since \( \frac{4}{0} \) is undefined, the planes are not parallel.

- **Calculate the Angle**:
\[ |\vec{n_1}| = \sqrt{4^2 + 8^2 + 1^2} = \sqrt{16 + 64 + 1} = \sqrt{81} = 9 \]
\[ |\vec{n_2}| = \sqrt{0^2 + 1^2 + 1^2} = \sqrt{0 + 1 + 1} = \sqrt{2} \]
\[ \cos \theta = \frac{9}{9 \cdot \sqrt{2}} = \frac{1}{\sqrt{2}} \]
\[ \theta = 45^\circ \]

### Summary of Results:
1. **(i)** Not parallel, not perpendicular, angle \( \theta = \cos^{-1}(-\frac{2}{5}) \)
2. **(ii)** Perpendicular
3. **(iii)** Parallel
4. **(iv)** Parallel
5. **(v)** Not parallel, not perpendicular, angle \( \theta = 45^\circ \)