Given that `veca`and `vecb` are unit vectors. If the vectors `vecp=3veca-5vecb` and `vecq=veca+vecb` are mutually perpendicular, then

Options

A. `veca` and `vecb` are also mutually perpendicular
B. `veca` and `vecb` have opposite directions
C. `veca` and `vecb` make an angle of `pi/3`
D. none of these

Explanation

To solve the problem, we need to analyze the given vectors and their properties. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Given Vectors**:
We are given two unit vectors, \(\vec{a}\) and \(\vec{b}\). The vectors \(\vec{p}\) and \(\vec{q}\) are defined as:
\[ \vec{p} = 3\vec{a} - 5\vec{b} \]
\[ \vec{q} = \vec{a} + \vec{b} \]

2. **Condition of Perpendicularity**:
Since \(\vec{p}\) and \(\vec{q}\) are mutually perpendicular, their dot product must be zero:
\[ \vec{p} \cdot \vec{q} = 0 \]

3. **Calculating the Dot Product**:
Substitute the expressions for \(\vec{p}\) and \(\vec{q}\):
\[ (3\vec{a} - 5\vec{b}) \cdot (\vec{a} + \vec{b}) = 0 \]
Expanding this using the distributive property of the dot product:
\[ 3\vec{a} \cdot \vec{a} + 3\vec{a} \cdot \vec{b} - 5\vec{b} \cdot \vec{a} - 5\vec{b} \cdot \vec{b} = 0 \]

4. **Using Properties of Unit Vectors**:
Since \(\vec{a}\) and \(\vec{b}\) are unit vectors:
\[ \vec{a} \cdot \vec{a} = 1 \quad \text{and} \quad \vec{b} \cdot \vec{b} = 1 \]
Thus, we can simplify the equation:
\[ 3(1) + 3(\vec{a} \cdot \vec{b}) - 5(\vec{a} \cdot \vec{b}) - 5(1) = 0 \]
This simplifies to:
\[ 3 + 3(\vec{a} \cdot \vec{b}) - 5(\vec{a} \cdot \vec{b}) - 5 = 0 \]
\[ 3 - 5 + (3 - 5)(\vec{a} \cdot \vec{b}) = 0 \]
\[ -2 + (-2)(\vec{a} \cdot \vec{b}) = 0 \]

5. **Solving for \(\vec{a} \cdot \vec{b}\)**:
Rearranging gives:
\[ -2(\vec{a} \cdot \vec{b}) = 2 \]
\[ \vec{a} \cdot \vec{b} = -1 \]

6. **Interpreting the Result**:
The dot product of two vectors is given by:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
Since both \(\vec{a}\) and \(\vec{b}\) are unit vectors, we have:
\[ \cos \theta = -1 \]
This implies that:
\[ \theta = 180^\circ \]
Therefore, \(\vec{a}\) and \(\vec{b}\) are in opposite directions.

### Final Conclusion:
The vectors \(\vec{a}\) and \(\vec{b}\) are opposite to each other.