The vector `vecP` makes `120^@` with the x-axis and vector Q  makes 30° with the y-axis. What is their resultant?

Options

A. `P+Q`
B. `P-Q`
C. `sqrt(P^(2)+Q^(2))`
D. `sqrt(P^(2)-Q^(2))`

Answer

To find the resultant of the vectors \( \vec{P} \) and \( \vec{Q} \), we will follow these steps:

### Step 1: Understand the Angles
- Vector \( \vec{P} \) makes an angle of \( 120^\circ \) with the x-axis.
- Vector \( \vec{Q} \) makes an angle of \( 30^\circ \) with the y-axis.

### Step 2: Determine the Angles with Respect to the X-Axis
- The angle \( \vec{P} \) makes with the x-axis is \( 120^\circ \).
- To find the angle \( \vec{Q} \) makes with the x-axis, we can calculate it as follows:
\[ \text{Angle of } \vec{Q} \text{ with x-axis} = 90^\circ - 30^\circ = 60^\circ \]

### Step 3: Express the Vectors in Component Form
- The components of vector \( \vec{P} \):
\[ \vec{P} = P \cos(120^\circ) \hat{i} + P \sin(120^\circ) \hat{j} \]
Using \( \cos(120^\circ) = -\frac{1}{2} \) and \( \sin(120^\circ) = \frac{\sqrt{3}}{2} \):
\[ \vec{P} = P \left(-\frac{1}{2}\right) \hat{i} + P \left(\frac{\sqrt{3}}{2}\right) \hat{j} = -\frac{P}{2} \hat{i} + \frac{P\sqrt{3}}{2} \hat{j} \]

- The components of vector \( \vec{Q} \):
\[ \vec{Q} = Q \cos(60^\circ) \hat{i} + Q \sin(60^\circ) \hat{j} \]
Using \( \cos(60^\circ) = \frac{1}{2} \) and \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \):
\[ \vec{Q} = Q \left(\frac{1}{2}\right) \hat{i} + Q \left(\frac{\sqrt{3}}{2}\right) \hat{j} = \frac{Q}{2} \hat{i} + \frac{Q\sqrt{3}}{2} \hat{j} \]

### Step 4: Add the Vectors
- The resultant vector \( \vec{R} = \vec{P} + \vec{Q} \):
\[ \vec{R} = \left(-\frac{P}{2} + \frac{Q}{2}\right) \hat{i} + \left(\frac{P\sqrt{3}}{2} + \frac{Q\sqrt{3}}{2}\right) \hat{j} \]
Simplifying gives:
\[ \vec{R} = \left(\frac{Q - P}{2}\right) \hat{i} + \left(\frac{(P + Q)\sqrt{3}}{2}\right) \hat{j} \]

### Step 5: Calculate the Magnitude of the Resultant
- The magnitude of the resultant vector \( R \) is given by:
\[ R = \sqrt{\left(\frac{Q - P}{2}\right)^2 + \left(\frac{(P + Q)\sqrt{3}}{2}\right)^2} \]
Simplifying:
\[ R = \frac{1}{2} \sqrt{(Q - P)^2 + 3(P + Q)^2} \]

### Step 6: Conclusion
- The resultant vector \( \vec{R} \) can be expressed in terms of its components or its magnitude, depending on the values of \( P \) and \( Q \).