Let ` vec a= hat i- hat j ,\ vec b=3 hat j- hat k\ a n d\ vec c=7 hat i- hat kdot\ ` Find a vector ` vec d` which is perpendicular to both ` vec a\ a n d\ vec b ,\ a n d\ vec cdot vec d=1.`

Explanation

To find the vector \( \vec{d} \) that is perpendicular to both \( \vec{a} \) and \( \vec{b} \) and satisfies the condition \( \vec{c} \cdot \vec{d} = 1 \), we can follow these steps:

### Step 1: Define the vectors
Given:
\[ \vec{a} = \hat{i} - \hat{j} \]
\[
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