Two vectors a and b have equal magnitudes of 12 units. These vectors are making angles 30°and 120° with the x axis respectively. Their sum is r . Find the x and y components of r .

Options

A. `(6sqrt(3) -6), (6-6sqrt(3))`
B. `(6sqrt(3) +6), (6+6sqrt(3))`
C. `(6sqrt(3)-6), (6+6sqrt(3))`
D. `(6sqrt(3) +6), (6-6sqrt(3))`

Answer

To find the x and y components of the resultant vector \( \mathbf{r} \) formed by the addition of two vectors \( \mathbf{a} \) and \( \mathbf{b} \), we can follow these steps:

### Step 1: Identify the vectors and their components
Given:
- Magnitude of both vectors \( |\mathbf{a}| = |\mathbf{b}| = 12 \) units.
- Angle of vector \( \mathbf{a} \) with the x-axis \( \theta_a = 30^\circ \).
- Angle of vector \( \mathbf{b} \) with the x-axis \( \theta_b = 120^\circ \).

### Step 2: Calculate the x and y components of vector \( \mathbf{a} \)
The x and y components of vector \( \mathbf{a} \) can be calculated using the following formulas:
- \( a_x = |\mathbf{a}| \cos(\theta_a) \)
- \( a_y = |\mathbf{a}| \sin(\theta_a) \)

Substituting the values:
- \( a_x = 12 \cos(30^\circ) = 12 \cdot \frac{\sqrt{3}}{2} = 6\sqrt{3} \)
- \( a_y = 12 \sin(30^\circ) = 12 \cdot \frac{1}{2} = 6 \)

Thus, the components of vector \( \mathbf{a} \) are:
\[ \mathbf{a} = 6\sqrt{3} \hat{i} + 6 \hat{j} \]

### Step 3: Calculate the x and y components of vector \( \mathbf{b} \)
Similarly, for vector \( \mathbf{b} \):
- \( b_x = |\mathbf{b}| \cos(\theta_b) \)
- \( b_y = |\mathbf{b}| \sin(\theta_b) \)

Substituting the values:
- \( b_x = 12 \cos(120^\circ) = 12 \cdot \left(-\frac{1}{2}\right) = -6 \)
- \( b_y = 12 \sin(120^\circ) = 12 \cdot \frac{\sqrt{3}}{2} = 6\sqrt{3} \)

Thus, the components of vector \( \mathbf{b} \) are:
\[ \mathbf{b} = -6 \hat{i} + 6\sqrt{3} \hat{j} \]

### Step 4: Find the resultant vector \( \mathbf{r} \)
The resultant vector \( \mathbf{r} \) is the sum of vectors \( \mathbf{a} \) and \( \mathbf{b} \):
\[ \mathbf{r} = \mathbf{a} + \mathbf{b} \]

Calculating the x and y components:
- \( r_x = a_x + b_x = 6\sqrt{3} - 6 \)
- \( r_y = a_y + b_y = 6 + 6\sqrt{3} \)

Thus, the resultant vector \( \mathbf{r} \) can be expressed as:
\[ \mathbf{r} = (6\sqrt{3} - 6) \hat{i} + (6 + 6\sqrt{3}) \hat{j} \]

### Step 5: Final Result
The x and y components of the resultant vector \( \mathbf{r} \) are:
- \( r_x = 6\sqrt{3} - 6 \)
- \( r_y = 6 + 6\sqrt{3} \)