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

Explanation

To solve the problem, we need to find a vector \( \vec{d} \) that is perpendicular to both vectors \( \vec{a} \) and \( \vec{b} \), and also satisfies the condition \( \vec{c} \cdot \vec{d} = 15 \).

### 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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