If ` vec adot hat i= vec adot( hat i+ hat j)= vec adot( hat i+ hat j+ hat k)=1,\ t h e n\ vec a=`
a.` vec0` b. ` hat i` c. ` hat j` d. ` hat i+ hat j+ hat k`

Explanation

To solve the problem, we need to analyze the given equations involving the vector \( \vec{a} \) and the unit vectors \( \hat{i}, \hat{j}, \hat{k} \).

### Step-by-Step Solution:

1. **Understanding the Dot Product**:
The dot product of two vectors \( \vec{a} \) and \( \hat{i} \) is given by:
\[
\vec{a} \cdot \hat{i} = A_1
...