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 \( \vec{c} \cdot \vec{d} = 15 \), we will follow these steps:

### Step 1: Define the vectors
Given:
- \( \vec{a} = \hat{i} + 4\hat{j} + 2\hat{k} \)
- \( \vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k} \)
- \( \vec{c} = 2\hat{i} - \hat{j} + 4\hat{k} \)

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