The value of `|vecA+vecB-vecC+vecD|` can be zero if:-

Options

A. `|vecA|=5,|vecB|=3,|vecC|=4,|vecD|=12`
B. `|vecA|=2sqrt(2),|vecB|=2,|vecC|=2,|vecD|=5`
C. `|vecA|=2sqrt(2),|vecB|=2,|vecC|=2,|vecD|=10`
D. `|vecA|=5,|vecB|=4,|vecC|=3,|vecD|=8`

Answer

To determine when the expression \( |\vec{A} + \vec{B} - \vec{C} + \vec{D}| \) can be zero, we need to analyze the conditions under which the vector sum results in the zero vector.

### Step-by-step Solution:

1. **Understanding Modulus of a Vector**:
The modulus (or magnitude) of a vector \(\vec{V}\) is denoted as \(|\vec{V}|\) and represents its length. For the expression to equal zero, the vector itself must be the zero vector.

2. **Setting Up the Equation**:
We need to find conditions under which:
\[ \vec{A} + \vec{B} - \vec{C} + \vec{D} = \vec{0} \]
This implies:
\[ \vec{A} + \vec{B} + \vec{D} = \vec{C} \]

3. **Using Unit Vectors**:
Let's express each vector in terms of unit vectors and their magnitudes:
- \(\vec{A} = |\vec{A}| \hat{a}\)
- \(\vec{B} = |\vec{B}| \hat{b}\)
- \(\vec{C} = |\vec{C}| \hat{c}\)
- \(\vec{D} = |\vec{D}| \hat{d}\)

Here, \(\hat{a}, \hat{b}, \hat{c}, \hat{d}\) are the unit vectors in the direction of \(\vec{A}, \vec{B}, \vec{C}, \vec{D}\) respectively.

4. **Substituting into the Equation**:
Substitute the expressions for the vectors into the equation:
\[ |\vec{A}| \hat{a} + |\vec{B}| \hat{b} + |\vec{D}| \hat{d} = |\vec{C}| \hat{c} \]

5. **Assuming Directions**:
To simplify, assume:
- \(\hat{a} = \hat{b} = -\hat{d}\)
- \(\hat{c} = \hat{d}\)

This means all vectors are in the same or opposite directions.

6. **Rearranging the Equation**:
The equation becomes:
\[ |\vec{A}| + |\vec{B}| - |\vec{C}| + |\vec{D}| = 0 \]

7. **Testing the Options**:
Now, we will test the provided options to see which one satisfies this equation.

- **Option A**: \( |\vec{A}| = 5, |\vec{B}| = 3, |\vec{C}| = 4, |\vec{D}| = 12 \)
\[ 5 + 3 - 4 + 12 = 16 \quad (\text{Not zero}) \]
- **Option B**: \( |\vec{A}| = 2\sqrt{2}, |\vec{B}| = 2, |\vec{C}| = 2, |\vec{D}| = 5 \)
\[ 2\sqrt{2} + 2 - 2 + 5 = 2\sqrt{2} + 5 \quad (\text{Not zero}) \]
- **Option C**: \( |\vec{A}| = 2\sqrt{2}, |\vec{B}| = 2, |\vec{C}| = 2, |\vec{D}| = 10 \)
\[ 2\sqrt{2} + 2 - 2 + 10 = 2\sqrt{2} + 10 \quad (\text{Not zero}) \]
- **Option D**: \( |\vec{A}| = 5, |\vec{B}| = 4, |\vec{C}| = 3, |\vec{D}| = 8 \)
\[ 5 + 4 - 3 + 8 = 14 \quad (\text{Not zero}) \]

8. **Conclusion**:
After checking all options, it appears that none of the options satisfy the condition for the modulus to be zero.