If non-zero vectors `veca and vecb` are equally inclined to coplanar vector `vecc`, then `vecc` can be

Options

A. `(|a|)/(|a|=2|b|)a+(|b|)/(|a|+|b|)b`
B. `|b|/(|a|+|b|)a+|a|/(|a|+|b|)b`
C. `(|a|)/(|a|+|b|)a+(|b|)/(|a|+2|b|)b`
D. `(|b|)/(2|a|+|b|)a+(|a|)/(2|a|+|b|)b`

Explanation

To solve the problem, we need to find the expression for the vector \(\vec{c}\) when the non-zero vectors \(\vec{a}\) and \(\vec{b}\) are equally inclined to the coplanar vector \(\vec{c}\).

### Step-by-step Solution:

1. **Understanding the Condition of Equal Inclination**:
- Two vectors \(\vec{a}\) and \(\vec{b}\) are said to be equally inclined to a vector \(\vec{c}\) if the angle between \(\vec{a}\) and \(\vec{c}\) is equal to the angle between \(\vec{b}\) and \(\vec{c}\). This can be mathematically expressed using the dot product.

2. **Using the Dot Product**:
- The condition for equal inclination can be expressed as:
\[ \frac{\vec{a} \cdot \vec{c}}{|\vec{a}| |\vec{c}|} = \frac{\vec{b} \cdot \vec{c}}{|\vec{b}| |\vec{c}|} \]
- This implies:
\[ \vec{a} \cdot \vec{c} |\vec{b}| = \vec{b} \cdot \vec{c} |\vec{a}| \]

3. **Expressing \(\vec{c}\)**:
- Since \(\vec{c}\) is coplanar with \(\vec{a}\) and \(\vec{b}\), we can express \(\vec{c}\) as a linear combination of \(\vec{a}\) and \(\vec{b}\):
\[ \vec{c} = k_1 \vec{a} + k_2 \vec{b} \]
- Here, \(k_1\) and \(k_2\) are scalars.

4. **Finding the Scalars**:
- To satisfy the condition of equal inclination, we can set:
\[ k_1 = \frac{|\vec{b}|}{|\vec{a}| + |\vec{b}|}, \quad k_2 = \frac{|\vec{a}|}{|\vec{a}| + |\vec{b}|} \]
- Therefore, we can write:
\[ \vec{c} = \frac{|\vec{b}|}{|\vec{a}| + |\vec{b}|} \vec{a} + \frac{|\vec{a}|}{|\vec{a}| + |\vec{b}|} \vec{b} \]

5. **Final Expression for \(\vec{c}\)**:
- Combining the above, we have:
\[ \vec{c} = \frac{|\vec{b}| \vec{a} + |\vec{a}| \vec{b}}{|\vec{a}| + |\vec{b}|} \]

### Conclusion:
Thus, the vector \(\vec{c}\) can be expressed as:
\[ \vec{c} = \frac{|\vec{b}| \vec{a} + |\vec{a}| \vec{b}}{|\vec{a}| + |\vec{b}|} \]

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