The sine of the angle between the line `(x-2)/(3) = (y-3)/(4) = (z-4)/(5)` and the plane `2x-2y+z=5` is

Options

A. `(10)/(6 sqrt5)`
B. `(4)/( 5 sqrt2)`
C. `(2 sqrt3)/(5)`
D. `( sqrt2)/(10)`

Answer

To find the sine of the angle between the given line and the plane, we can follow these steps:

### Step 1: Identify the direction ratios of the line and the normal to the plane.
The line is given by the equation:
\[ \frac{x-2}{3} = \frac{y-3}{4} = \frac{z-4}{5} \]
From this, we can extract the direction ratios of the line, which are \( a = 3, b = 4, c = 5 \).

The equation of the plane is:
\[ 2x - 2y + z = 5 \]
From this equation, we can identify the normal vector to the plane, which has direction ratios \( A = 2, B = -2, C = 1 \).

### Step 2: Calculate the dot product of the direction ratios of the line and the normal to the plane.
The dot product \( \vec{A} \cdot \vec{B} \) is calculated as follows:
\[ \vec{A} \cdot \vec{B} = (3)(2) + (4)(-2) + (5)(1) = 6 - 8 + 5 = 3 \]

### Step 3: Calculate the magnitudes of the direction ratios.
The magnitude of the normal vector \( \vec{A} \) is:
\[ |\vec{A}| = \sqrt{2^2 + (-2)^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

The magnitude of the direction ratios of the line \( \vec{B} \) is:
\[ |\vec{B}| = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} = 5\sqrt{2} \]

### Step 4: Use the dot product to find \( \cos \phi \).
Using the formula for the cosine of the angle \( \phi \) between the line and the normal to the plane:
\[ \cos \phi = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|} = \frac{3}{3 \cdot 5\sqrt{2}} = \frac{1}{5\sqrt{2}} \]

### Step 5: Relate \( \phi \) to \( \theta \).
Since the angle \( \theta \) is the angle between the line and the plane, we know that:
\[ \phi + \theta = 90^\circ \implies \theta = 90^\circ - \phi \]

### Step 6: Use the sine-cosine relationship.
From trigonometric identities, we have:
\[ \sin \theta = \cos \phi \]
Thus,
\[ \sin \theta = \frac{1}{5\sqrt{2}} \]

### Final Result
The sine of the angle between the line and the plane is:
\[ \sin \theta = \frac{1}{5\sqrt{2}} = \frac{\sqrt{2}}{10} \]