Determine that vector which when added to the resultant of `vec(A)=3hati-5hatj+7hatk` and `vec(B)=2hati+4hatj-3hatk` gives unit vector along y- direction.

Explanation

To solve the problem, we need to find a vector \( \vec{X} \) that, when added to the resultant of vectors \( \vec{A} \) and \( \vec{B} \), gives the unit vector along the y-direction, which is represented as \( \hat{j} \).

### Step 1: Find the resultant vector \( \vec{R} \) of \( \vec{A} \) and \( \vec{B} \)

Given:
\[ \vec{A} = 3\hat{i} - 5\hat{j} + 7\hat{k} \]
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