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

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