Let ` vec a= hat i+4 hat j+2 hat k ,\ \ vec b=3 hat i-\ 2 hat j+7 hat k` and ` vec c=2 hat i-\ hat j+4 hat k` . Find a vector ` vec d` which is perpendicular to both ` vec a` and ` vec b` and ` vec c` . ` vec d` =15.

Explanation

To find a vector \(\vec{d}\) that is perpendicular to both \(\vec{a}\) and \(\vec{b}\), and satisfies \(\vec{c} \cdot \vec{d} = 15\), we can follow these steps:

### Step 1: Define the vectors
Given:
\[ \vec{a} = \hat{i} + 4\hat{j} + 2\hat{k} \]
\[
...