If `bara and barb` are two unit vectors at `120^@` and c is any vector inclined to `bara and barb` at an angle `theta`. The complete set of values of `cos theta ` is

Options

A. `[-1/2,1/2]`
B. `[0,1/2]`
C. `[1/2,sqrt3/2]`
D. `[-sqrt3/2,sqrt3/2]`

Answer

To solve the problem step by step, we will analyze the given information and derive the complete set of values for \( \cos \theta \).

### Step 1: Understand the vectors
We have two unit vectors \( \mathbf{a} \) and \( \mathbf{b} \) that are inclined at an angle of \( 120^\circ \) to each other. Since they are unit vectors, we have:
\[ |\mathbf{a}| = |\mathbf{b}| = 1 \]

### Step 2: Set up the angle relationships
Let \( \theta \) be the angle between vector \( \mathbf{c} \) and vector \( \mathbf{a} \). Since \( \mathbf{c} \) is also inclined to vector \( \mathbf{b} \) at the same angle \( \theta \), the angle between \( \mathbf{c} \) and \( \mathbf{b} \) will also be \( \theta \).

### Step 3: Use the angle between \( \mathbf{a} \) and \( \mathbf{b} \)
The angle between vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given as \( 120^\circ \). Therefore, the angle between \( \mathbf{c} \) and \( \mathbf{b} \) can be expressed as:
\[ \text{Angle between } \mathbf{a} \text{ and } \mathbf{b} = \theta + \theta = 2\theta \]
Thus, we have:
\[ 2\theta = 120^\circ \]

### Step 4: Solve for \( \theta \)
Now, we can solve for \( \theta \):
\[ \theta = \frac{120^\circ}{2} = 60^\circ \]

### Step 5: Consider the negative angle
Since vector \( \mathbf{c} \) can also be in the opposite direction, we also consider:
\[ \theta = -60^\circ \]

### Step 6: Calculate \( \cos \theta \)
Now we calculate \( \cos \theta \) for both values:
1. For \( \theta = 60^\circ \):
\[ \cos(60^\circ) = \frac{1}{2} \]
2. For \( \theta = -60^\circ \):
\[ \cos(-60^\circ) = \frac{1}{2} \]

### Step 7: Determine the complete set of values
Since \( \mathbf{c} \) can be in both directions, the complete set of values for \( \cos \theta \) will range from:
\[ \cos \theta \in \left[-\frac{1}{2}, \frac{1}{2}\right] \]

### Final Answer
The complete set of values of \( \cos \theta \) is:
\[ \cos \theta \in \left[-\frac{1}{2}, \frac{1}{2}\right] \]

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