Let `veca, vecb and vecc` be the three vectors having magnitudes, 1,5 and 3, respectively, such that the angle between `veca and vecb "is" theta and veca xx (veca xxvecb)=vecc`. Then `tan theta` is equal to

Options

A. 0
B. `2/3`
C. `3/5`
D. `3/4`

Explanation

To solve the problem, we need to find the value of \( \tan \theta \) given the vectors \( \vec{a}, \vec{b}, \vec{c} \) with magnitudes 1, 5, and 3 respectively, and the condition \( \vec{a} \times (\vec{a} \times \vec{b}) = \vec{c} \).

### Step-by-Step Solution:

1. **Understanding the Cross Product**:
The expression \( \vec{a} \times (\vec{a} \times \vec{b}) \) can be simplified using the vector triple product identity:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]
In our case, it simplifies to:
\[ \vec{a} \times (\vec{a} \times \vec{b}) = (\vec{a} \cdot \vec{b}) \vec{a} - (\vec{a} \cdot \vec{a}) \vec{b} \]

2. **Magnitude of Vectors**:
Given the magnitudes:
\[ |\vec{a}| = 1, \quad |\vec{b}| = 5, \quad |\vec{c}| = 3 \]

3. **Finding the Dot Product**:
The dot product \( \vec{a} \cdot \vec{b} \) can be expressed as:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta = 1 \cdot 5 \cdot \cos \theta = 5 \cos \theta \]

4. **Substituting into the Equation**:
We can substitute back into our equation:
\[ \vec{a} \times (\vec{a} \times \vec{b}) = (5 \cos \theta) \vec{a} - (1) \vec{b} = \vec{c} \]
This gives us:
\[ (5 \cos \theta) \vec{a} - 5 \vec{b} = \vec{c} \]

5. **Finding Magnitude**:
Taking magnitudes on both sides:
\[ |(5 \cos \theta) \vec{a} - 5 \vec{b}| = |\vec{c}| \]
This leads to:
\[ |(5 \cos \theta) \vec{a} - 5 \vec{b}| = 3 \]

6. **Using the Pythagorean Theorem**:
To find \( \tan \theta \), we can use the sine and cosine relationship:
From the sine definition:
\[ \sin \theta = \frac{3}{5} \]
Using the Pythagorean identity:
\[ \cos^2 \theta + \sin^2 \theta = 1 \implies \cos^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \]
Thus,
\[ \cos \theta = \frac{4}{5} \]

7. **Calculating \( \tan \theta \)**:
Now we can find \( \tan \theta \):
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \]

### Final Answer:
Thus, the value of \( \tan \theta \) is:
\[ \boxed{\frac{3}{4}} \]