If `hata and hatb` are two unit vectors and `theta` is the angle between them, then `hata+ hatb` is a unit vector, if

Options

A. `theta = pi/3`
B. `theta = pi/4`
C. `theta = pi/2`
D. `theta = (2pi)/3`

Answer

To determine the condition under which the sum of two unit vectors \(\hat{a}\) and \(\hat{b}\) is also a unit vector, we can follow these steps:

### Step 1: Understand the vectors
Let \(\hat{a}\) and \(\hat{b}\) be two unit vectors. By definition, the magnitude of each unit vector is 1:
\[ |\hat{a}| = 1 \quad \text{and} \quad |\hat{b}| = 1 \]

### Step 2: Write the expression for the sum of the vectors
We want to find the condition under which the vector sum \(\hat{a} + \hat{b}\) is a unit vector. This means:
\[ |\hat{a} + \hat{b}| = 1 \]

### Step 3: Square both sides
Squaring both sides gives:
\[ |\hat{a} + \hat{b}|^2 = 1^2 \]
This expands to:
\[ (\hat{a} + \hat{b}) \cdot (\hat{a} + \hat{b}) = 1 \]

### Step 4: Use the properties of dot product
Using the distributive property of the dot product, we can expand this:
\[ \hat{a} \cdot \hat{a} + \hat{b} \cdot \hat{b} + 2 \hat{a} \cdot \hat{b} = 1 \]
Since \(\hat{a}\) and \(\hat{b}\) are unit vectors:
\[ 1 + 1 + 2 \hat{a} \cdot \hat{b} = 1 \]
This simplifies to:
\[ 2 + 2 \hat{a} \cdot \hat{b} = 1 \]

### Step 5: Solve for the dot product
Rearranging gives:
\[ 2 \hat{a} \cdot \hat{b} = 1 - 2 \]
\[ 2 \hat{a} \cdot \hat{b} = -1 \]
Thus:
\[ \hat{a} \cdot \hat{b} = -\frac{1}{2} \]

### Step 6: Relate the dot product to the angle
The dot product of two vectors can also be expressed in terms of the angle \(\theta\) between them:
\[ \hat{a} \cdot \hat{b} = |\hat{a}| |\hat{b}| \cos \theta = 1 \cdot 1 \cdot \cos \theta = \cos \theta \]
Therefore, we have:
\[ \cos \theta = -\frac{1}{2} \]

### Step 7: Find the angle \(\theta\)
The angle \(\theta\) for which \(\cos \theta = -\frac{1}{2}\) corresponds to:
\[ \theta = \frac{2\pi}{3} \quad \text{or} \quad \theta = \frac{4\pi}{3} \quad \text{(in radians)} \]

### Conclusion
Thus, the vectors \(\hat{a}\) and \(\hat{b}\) will sum to a unit vector if the angle \(\theta\) between them is either \(\frac{2\pi}{3}\) or \(\frac{4\pi}{3}\).

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