If `P+Q=R and |P|=|Q|= sqrt(3) and |R| ==3 ,` then the angle between P and Q is

Options

A. `pi //4`
B. `pi//6`
C. `pi//3`
D. `pi//2`

Explanation

To solve the problem, we need to find the angle between vectors \( P \) and \( Q \) given the following conditions:

1. \( P + Q = R \)
2. \( |P| = |Q| = \sqrt{3} \)
3. \( |R| = 3 \)

### Step-by-Step Solution:

**Step 1: Write the equation for the magnitude of the resultant vector.**

From vector addition, we know that:
\[ |R| = |P + Q| = \sqrt{|P|^2 + |Q|^2 + 2|P||Q|\cos\theta} \]
where \( \theta \) is the angle between vectors \( P \) and \( Q \).

**Step 2: Substitute the known magnitudes into the equation.**

We know that:
- \( |P| = \sqrt{3} \)
- \( |Q| = \sqrt{3} \)
- \( |R| = 3 \)

Substituting these values into the equation gives:
\[ 3 = \sqrt{(\sqrt{3})^2 + (\sqrt{3})^2 + 2(\sqrt{3})(\sqrt{3})\cos\theta} \]

**Step 3: Simplify the equation.**

Calculating the squares:
\[ 3 = \sqrt{3 + 3 + 6\cos\theta} \]
This simplifies to:
\[ 3 = \sqrt{6 + 6\cos\theta} \]

**Step 4: Square both sides to eliminate the square root.**

Squaring both sides results in:
\[ 9 = 6 + 6\cos\theta \]

**Step 5: Solve for \( \cos\theta \).**

Rearranging the equation gives:
\[ 9 - 6 = 6\cos\theta \]
\[ 3 = 6\cos\theta \]
\[ \cos\theta = \frac{3}{6} = \frac{1}{2} \]

**Step 6: Find the angle \( \theta \).**

The angle \( \theta \) for which \( \cos\theta = \frac{1}{2} \) is:
\[ \theta = 60^\circ \quad \text{or} \quad \theta = \frac{\pi}{3} \text{ radians} \]

### Final Answer:
The angle between vectors \( P \) and \( Q \) is \( \frac{\pi}{3} \) radians.