In x-y plane, a force `10 N` acts at an angle `30^(@)` to the positive direction of x-axis. The force can be written as

Options

A. `5hat(i) + 5hat(j)`
B. `5sqrt(3)hat(i) + 5hat(j) N`
C. `5hat(i) + 5sqrt(3) hat(j) N`
D. None of these

Answer

To express the force of 10 N acting at an angle of 30° to the positive x-axis in vector form, we can follow these steps:

### Step 1: Identify the components of the force
The force can be resolved into two components: one along the x-axis and one along the y-axis. The formulas for the components are:
- \( F_x = F \cdot \cos(\theta) \)
- \( F_y = F \cdot \sin(\theta) \)

Where:
- \( F \) is the magnitude of the force (10 N)
- \( \theta \) is the angle (30°)

### Step 2: Calculate the x-component
Using the formula for the x-component:
\[ F_x = 10 \cdot \cos(30^\circ) \]
We know that \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \):
\[ F_x = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3} \text{ N} \]

### Step 3: Calculate the y-component
Using the formula for the y-component:
\[ F_y = 10 \cdot \sin(30^\circ) \]
We know that \( \sin(30^\circ) = \frac{1}{2} \):
\[ F_y = 10 \cdot \frac{1}{2} = 5 \text{ N} \]

### Step 4: Write the force vector in component form
The force vector \( \vec{F} \) can be expressed in terms of its components along the x and y axes:
\[ \vec{F} = F_x \hat{i} + F_y \hat{j} \]
Substituting the values we calculated:
\[ \vec{F} = 5\sqrt{3} \hat{i} + 5 \hat{j} \text{ N} \]

### Final Answer
Thus, the force can be written as:
\[ \vec{F} = 5\sqrt{3} \hat{i} + 5 \hat{j} \text{ N} \]

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