The angle between the planes 3x-4y+5z=0 and 2x-y-cz=5 is

Options

A. `(pi)/(3)`
B. `(pi)/(2)`
C. `(pi)/(6)`
D. `(pi)/(4)`

Answer

To find the angle between the planes given by the equations \(3x - 4y + 5z = 0\) and \(2x - y - 2z = 5\), we can follow these steps:

### Step 1: Identify the normal vectors of the planes
The normal vector of a plane given by the equation \(Ax + By + Cz = D\) is \(\langle A, B, C \rangle\).

- For the first plane \(3x - 4y + 5z = 0\), the coefficients give us the normal vector:
\[ \mathbf{n_1} = \langle 3, -4, 5 \rangle \]

- For the second plane \(2x - y - 2z = 5\), the coefficients give us the normal vector:
\[ \mathbf{n_2} = \langle 2, -1, -2 \rangle \]

### Step 2: Use the formula for the angle between two planes
The angle \(\theta\) between two planes can be found using the formula:
\[ \cos \theta = \frac{\mathbf{n_1} \cdot \mathbf{n_2}}{|\mathbf{n_1}| |\mathbf{n_2}|} \]

### Step 3: Calculate the dot product \(\mathbf{n_1} \cdot \mathbf{n_2}\)
The dot product is calculated as follows:
\[ \mathbf{n_1} \cdot \mathbf{n_2} = (3)(2) + (-4)(-1) + (5)(-2) \]
Calculating each term:
- \(3 \cdot 2 = 6\)
- \(-4 \cdot -1 = 4\)
- \(5 \cdot -2 = -10\)

Thus,
\[ \mathbf{n_1} \cdot \mathbf{n_2} = 6 + 4 - 10 = 0 \]

### Step 4: Calculate the magnitudes of \(\mathbf{n_1}\) and \(\mathbf{n_2}\)
The magnitude of a vector \(\langle a, b, c \rangle\) is given by \(\sqrt{a^2 + b^2 + c^2}\).

- For \(\mathbf{n_1}\):
\[ |\mathbf{n_1}| = \sqrt{3^2 + (-4)^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \]

- For \(\mathbf{n_2}\):
\[ |\mathbf{n_2}| = \sqrt{2^2 + (-1)^2 + (-2)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

### Step 5: Substitute values into the cosine formula
Now substituting the values into the cosine formula:
\[ \cos \theta = \frac{0}{|\mathbf{n_1}| |\mathbf{n_2}|} = \frac{0}{\sqrt{50} \cdot 3} = 0 \]

### Step 6: Determine the angle \(\theta\)
Since \(\cos \theta = 0\), this implies:
\[ \theta = \frac{\pi}{2} \]

### Conclusion
The angle between the planes \(3x - 4y + 5z = 0\) and \(2x - y - 2z = 5\) is \(\frac{\pi}{2}\).