If u, v, w are unit vectors satisfying `2u+2v+2w=0," then "abs(u-v)` equals

Options

A. `7/4`
B. `sqrt(5/2)`
C. `sqrt(7/2)`
D. `5/4`

Answer

To solve the problem, we need to find the value of |u - v| given that u, v, and w are unit vectors satisfying the equation:

\[ 2u + 2v + 2w = 0 \]

### Step-by-Step Solution:

1. **Simplify the Given Equation**:
Start by dividing the entire equation by 2:
\[ u + v + w = 0 \]
Rearranging gives:
\[ u + v = -w \]

**Hint**: Remember that unit vectors have a magnitude of 1.

2. **Find the Magnitude**:
Now, take the magnitude of both sides:
\[ |u + v| = |-w| = |w| = 1 \]

**Hint**: Use the property of magnitudes: |a + b|² = |a|² + |b|² + 2a·b.

3. **Square the Magnitude**:
Squaring both sides gives:
\[ |u + v|^2 = 1^2 = 1 \]
Expanding the left side:
\[ |u|^2 + |v|^2 + 2u·v = 1 \]
Since u and v are unit vectors:
\[ 1 + 1 + 2u·v = 1 \]
This simplifies to:
\[ 2 + 2u·v = 1 \]
Thus:
\[ 2u·v = 1 - 2 \implies u·v = -\frac{1}{2} \]

**Hint**: The dot product of two vectors can give insight into the angle between them.

4. **Find |u - v|**:
Now, we need to find |u - v|. We can use the formula:
\[ |u - v|^2 = |u|^2 + |v|^2 - 2u·v \]
Substituting the known values:
\[ |u - v|^2 = 1 + 1 - 2(-\frac{1}{2}) \]
This simplifies to:
\[ |u - v|^2 = 2 + 1 = 3 \]

**Hint**: Remember to take the square root to find the magnitude.

5. **Final Calculation**:
Taking the square root gives:
\[ |u - v| = \sqrt{3} \]

### Conclusion:
Thus, the value of |u - v| is:
\[ \boxed{\sqrt{3}} \]