A displacement vector of magnitude 4 makes an angle `30^(@)` with the x-axis. Its rectangular components in x-y plane are

Options

A. `2sqrt3,2`
B. `4sqrt3,4`
C. `2/sqrt3,2`
D. `4/sqrt3,4`

Answer

To find the rectangular components of a displacement vector of magnitude 4 that makes an angle of 30 degrees with the x-axis, we can follow these steps:

### Step 1: Identify the given values
- Magnitude of the displacement vector (A) = 4
- Angle with the x-axis (θ) = 30 degrees

### Step 2: Use trigonometric functions to find components
The components of the vector can be found using the cosine and sine functions:
- The x-component (Ax) can be calculated using:
\[ Ax = A \cdot \cos(θ) \]
- The y-component (Ay) can be calculated using:
\[ Ay = A \cdot \sin(θ) \]

### Step 3: Calculate the x-component (Ax)
Substituting the values into the equation for Ax:
\[ Ax = 4 \cdot \cos(30^\circ) \]
Using the value of \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\):
\[ Ax = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \]

### Step 4: Calculate the y-component (Ay)
Substituting the values into the equation for Ay:
\[ Ay = 4 \cdot \sin(30^\circ) \]
Using the value of \(\sin(30^\circ) = \frac{1}{2}\):
\[ Ay = 4 \cdot \frac{1}{2} = 2 \]

### Step 5: Write the final answer
The rectangular components of the displacement vector are:
\[ Ax = 2\sqrt{3}, \quad Ay = 2 \]

Thus, the rectangular components in the x-y plane are \( (2\sqrt{3}, 2) \).