Let ` veca and vecb` are two vectors inclined at an angle of ` 60^(@) , If |veca|=|vecb|=2` ,the the angle between ` veca and veca + vecb` is

Options

A. ` 30^(@)`
B. `60^(@)`
C. `45^(@)`
D. none of these

Explanation

To solve the problem step by step, we need to find the angle between the vector \( \vec{a} \) and the vector \( \vec{a} + \vec{b} \) given that the magnitudes of both vectors are 2 and they are inclined at an angle of \( 60^\circ \).

### Step 1: Calculate \( \vec{a} \cdot \vec{b} \)

Given:
- \( |\vec{a}| = 2 \)
- \( |\vec{b}| = 2 \)
- \( \theta = 60^\circ \)

The dot product of two vectors can be calculated as:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \]

Substituting the values:
\[ \vec{a} \cdot \vec{b} = 2 \cdot 2 \cdot \cos(60^\circ) = 4 \cdot \frac{1}{2} = 2 \]

### Step 2: Calculate the magnitude of \( \vec{a} + \vec{b} \)

The magnitude of the sum of two vectors can be calculated using the formula:
\[ |\vec{a} + \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b}} \]

Substituting the known values:
\[ |\vec{a} + \vec{b}| = \sqrt{2^2 + 2^2 + 2 \cdot 2} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3} \]

### Step 3: Find the angle \( \phi \) between \( \vec{a} \) and \( \vec{a} + \vec{b} \)

The cosine of the angle \( \phi \) between \( \vec{a} \) and \( \vec{a} + \vec{b} \) can be calculated using the formula:
\[ \cos(\phi) = \frac{\vec{a} \cdot (\vec{a} + \vec{b})}{|\vec{a}| |\vec{a} + \vec{b}|} \]

First, calculate \( \vec{a} \cdot (\vec{a} + \vec{b}) \):
\[ \vec{a} \cdot (\vec{a} + \vec{b}) = \vec{a} \cdot \vec{a} + \vec{a} \cdot \vec{b} = |\vec{a}|^2 + \vec{a} \cdot \vec{b} = 2^2 + 2 = 4 + 2 = 6 \]

Now substituting into the cosine formula:
\[ \cos(\phi) = \frac{6}{2 \cdot 2\sqrt{3}} = \frac{6}{4\sqrt{3}} = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2} \]

### Step 4: Determine the angle \( \phi \)

From the cosine value:
\[ \cos(\phi) = \frac{\sqrt{3}}{2} \]

This corresponds to:
\[ \phi = 30^\circ \]

### Final Answer
The angle between \( \vec{a} \) and \( \vec{a} + \vec{b} \) is \( 30^\circ \).

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