Vector `vec(A)` is 2`cm` long and is `60^(@)` above the x-axis in the first quadrant. Vactor `vec(B)` is `2 cm` long and is `60^(@)` below the x-axis in the fourth quadrant. The sum `vec(A)+vec(B)` is a vector of magnitudes

Options

A. 2`cm` along positive y-axis
B. 2`cm` along positive x-axis
C. 2`cm` along negative y-axis
D. 2`cm` along negative x-axis

Explanation

To solve the problem of finding the magnitude of the resultant vector \( \vec{R} = \vec{A} + \vec{B} \), we will follow these steps:

### Step 1: Determine the components of vector \( \vec{A} \)
Vector \( \vec{A} \) has a magnitude of \( 2 \, \text{cm} \) and is directed \( 60^\circ \) above the x-axis. We can find its components using trigonometry:
- \( A_x = A \cos(60^\circ) = 2 \cos(60^\circ) = 2 \times \frac{1}{2} = 1 \, \text{cm} \)
- \( A_y = A \sin(60^\circ) = 2 \sin(60^\circ) = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3} \, \text{cm} \)

### Step 2: Determine the components of vector \( \vec{B} \)
Vector \( \vec{B} \) has a magnitude of \( 2 \, \text{cm} \) and is directed \( 60^\circ \) below the x-axis. Its components are:
- \( B_x = B \cos(60^\circ) = 2 \cos(60^\circ) = 2 \times \frac{1}{2} = 1 \, \text{cm} \)
- \( B_y = B \sin(-60^\circ) = 2 \sin(-60^\circ) = 2 \times -\frac{\sqrt{3}}{2} = -\sqrt{3} \, \text{cm} \)

### Step 3: Find the resultant components
Now, we can find the components of the resultant vector \( \vec{R} \):
- \( R_x = A_x + B_x = 1 + 1 = 2 \, \text{cm} \)
- \( R_y = A_y + B_y = \sqrt{3} + (-\sqrt{3}) = 0 \, \text{cm} \)

### Step 4: Calculate the magnitude of the resultant vector
The magnitude of the resultant vector \( \vec{R} \) is given by:
\[ |\vec{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2 \, \text{cm} \]

### Step 5: Determine the direction of the resultant vector
Since \( R_y = 0 \) and \( R_x = 2 \, \text{cm} \), the resultant vector \( \vec{R} \) is directed along the positive x-axis.

### Final Answer
The magnitude of the resultant vector \( \vec{R} \) is \( 2 \, \text{cm} \) and it is directed along the positive x-axis.

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