A plane is inclined at an angle `30^(@)` with horizontal. The component of a vector `vec(A)= -10k` perpendicular to this plane is: (here z-direction is vertically upwards)

Options

A. `5sqrt(2)`
B. `5sqrt(3)`e
C. `5`
D. `2.5`

Answer

To find the component of the vector \(\vec{A} = -10\hat{k}\) that is perpendicular to a plane inclined at an angle of \(30^\circ\) with the horizontal, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Vector Orientation**:
The vector \(\vec{A} = -10\hat{k}\) indicates that it is directed downwards along the z-axis (since \(\hat{k}\) is the unit vector in the z-direction).

2. **Identify the Plane's Orientation**:
The plane is inclined at \(30^\circ\) to the horizontal. This means that the normal (perpendicular) to the plane makes an angle of \(30^\circ\) with the vertical (z-axis).

3. **Determine the Normal Vector**:
The normal vector to the inclined plane can be represented as:
\[ \hat{n} = \sin(30^\circ)\hat{i} + \cos(30^\circ)\hat{k} \]
where \(\hat{i}\) is the horizontal direction (x-axis) and \(\hat{k}\) is the vertical direction (z-axis).

4. **Calculate the Components**:
- The sine and cosine values for \(30^\circ\) are:
\[ \sin(30^\circ) = \frac{1}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
- Thus, the normal vector becomes:
\[ \hat{n} = \frac{1}{2}\hat{i} + \frac{\sqrt{3}}{2}\hat{k} \]

5. **Find the Component of \(\vec{A}\) Perpendicular to the Plane**:
The component of the vector \(\vec{A}\) that is perpendicular to the plane can be calculated using the dot product:
\[ \text{Component} = \vec{A} \cdot \hat{n} \]
Substituting \(\vec{A} = -10\hat{k}\) and \(\hat{n}\):
\[ \text{Component} = (-10\hat{k}) \cdot \left(\frac{1}{2}\hat{i} + \frac{\sqrt{3}}{2}\hat{k}\right) \]
Since the dot product of \(\hat{i}\) and \(\hat{k}\) is zero, we only consider the \(\hat{k}\) component:
\[ \text{Component} = -10 \cdot \frac{\sqrt{3}}{2} = -5\sqrt{3} \]

6. **Final Result**:
The component of the vector \(\vec{A}\) that is perpendicular to the inclined plane is:
\[ -5\sqrt{3} \]