Let `vecu,vecv and vecw` be vectors such that `vecu+ vecv + vecw =0` if `|vecu|= 3, |vecv|=4 and |vecw|=5` then `vecu.vecv + vecv .vecw + vecw .vecu ` is

Options

A. `-25`
B. `-27`
C. `28`
D. `25`

Answer

To solve the problem, we need to find the value of \( \vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} \) given that \( \vec{u} + \vec{v} + \vec{w} = 0 \) and the magnitudes of the vectors are \( |\vec{u}| = 3 \), \( |\vec{v}| = 4 \), and \( |\vec{w}| = 5 \).

### Step-by-step Solution:

1. **Use the given equation:**
\[ \vec{u} + \vec{v} + \vec{w} = 0 \]
This implies that \( \vec{w} = -(\vec{u} + \vec{v}) \).

2. **Square both sides:**
Squaring the equation \( \vec{u} + \vec{v} + \vec{w} = 0 \) gives:
\[ |\vec{u} + \vec{v} + \vec{w}|^2 = 0 \]
Expanding this, we have:
\[ |\vec{u}|^2 + |\vec{v}|^2 + |\vec{w}|^2 + 2(\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u}) = 0 \]

3. **Substitute the magnitudes:**
We know:
\[ |\vec{u}|^2 = 3^2 = 9, \quad |\vec{v}|^2 = 4^2 = 16, \quad |\vec{w}|^2 = 5^2 = 25 \]
So substituting these values gives:
\[ 9 + 16 + 25 + 2(\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u}) = 0 \]

4. **Calculate the sum of squares:**
Adding the squares:
\[ 9 + 16 + 25 = 50 \]
Therefore, the equation becomes:
\[ 50 + 2(\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u}) = 0 \]

5. **Isolate the dot product terms:**
Rearranging gives:
\[ 2(\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u}) = -50 \]
Dividing by 2:
\[ \vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} = -25 \]

### Final Answer:
\[ \vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} = -25 \]