If `vec(m),vec(n),vec(r)` are three vectors, `theta` is the angle between the vectors `vec(m) and vec(n)`, what is mnr cos `theta` equal to ?

Options

A. `(vec(m).vec(n))(vec(r).(vec(r)//r))`
B. `(vec(m).vec(n))(vec(r).vec(r))`
C. `(vec(m).vec(r))(vec(n).(vec(n)//n))`
D. `(vec(m).vec(n))vec(r)`

Explanation

To solve the problem, we need to find the expression for \( m n r \cos \theta \) where \( \vec{m} \) and \( \vec{n} \) are two vectors and \( \theta \) is the angle between them.

### Step-by-Step Solution:

1. **Understanding the Dot Product**:
The cosine of the angle \( \theta \) between two vectors \( \vec{m} \) and \( \vec{n} \) can be expressed using the dot product:
\[ \cos \theta = \frac{\vec{m} \cdot \vec{n}}{|\vec{m}| |\vec{n}|} \]

2. **Rearranging the Equation**:
From the above equation, we can rearrange it to find \( \vec{m} \cdot \vec{n} \):
\[ \vec{m} \cdot \vec{n} = |\vec{m}| |\vec{n}| \cos \theta \]

3. **Substituting Values**:
Let \( |\vec{m}| = m \) and \( |\vec{n}| = n \). Therefore, we can write:
\[ \vec{m} \cdot \vec{n} = m n \cos \theta \]

4. **Incorporating the Third Vector**:
Now, we need to consider the third vector \( \vec{r} \). We want to find \( m n r \cos \theta \):
\[ m n r \cos \theta = (m n \cos \theta) r \]

5. **Final Expression**:
Since we have already established that \( \vec{m} \cdot \vec{n} = m n \cos \theta \), we can express \( m n r \cos \theta \) as:
\[ m n r \cos \theta = \vec{m} \cdot \vec{n} \cdot r \]

### Conclusion:
Thus, the expression \( m n r \cos \theta \) is equal to \( \vec{m} \cdot \vec{n} \cdot r \).