At what angle two vectors `vec(P) = 2N and vec(Q)= 3N ` act such that their sum is 4 N .

Explanation

To find the angle between two vectors \(\vec{P} = 2 \, \text{N}\) and \(\vec{Q} = 3 \, \text{N}\) such that their resultant is \(4 \, \text{N}\), we can use the formula for the resultant of two vectors:

\[ R = \sqrt{P^2 + Q^2 + 2PQ \cos \theta} \]

Where:
- \(R\) is the resultant vector,
- \(P\) and \(Q\) are the magnitudes of the vectors,
- \(\theta\) is the angle between the two vectors.

### Step 1: Write down the formula for the resultant of two vectors.
We start with the formula for the resultant of two vectors:

\[ R^2 = P^2 + Q^2 + 2PQ \cos \theta \]

### Step 2: Substitute the known values into the equation.
Given:
- \(R = 4 \, \text{N}\)
- \(P = 2 \, \text{N}\)
- \(Q = 3 \, \text{N}\)

Substituting these values into the equation:

\[ 4^2 = 2^2 + 3^2 + 2 \cdot 2 \cdot 3 \cdot \cos \theta \]

### Step 3: Simplify the equation.
Calculating the squares:

\[ 16 = 4 + 9 + 12 \cos \theta \]

Now combine the constants:

\[ 16 = 13 + 12 \cos \theta \]

### Step 4: Isolate the cosine term.
Subtract \(13\) from both sides:

\[ 16 - 13 = 12 \cos \theta \]

This simplifies to:

\[ 3 = 12 \cos \theta \]

### Step 5: Solve for \(\cos \theta\).
Divide both sides by \(12\):

\[ \cos \theta = \frac{3}{12} = \frac{1}{4} \]

### Step 6: Find the angle \(\theta\).
Now, take the inverse cosine:

\[ \theta = \cos^{-1}\left(\frac{1}{4}\right) \]

Using a calculator, we find:

\[ \theta \approx 75.52^\circ \]

### Final Answer:
The angle between the two vectors \(\vec{P}\) and \(\vec{Q}\) is approximately \(75.52^\circ\).

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