Find the value of `lambda` , which makes the vectors ` vec a ,\ vec b` and ` vec c` coplanar, where ` vec a=\ hat i+3 hat j+4 hat k ,\ \ vec b=2 hat i+lambda hat j+2 hat k` and ` vec c=4 hat i-7 hat j+10 hat k`

Explanation

To find the value of \( \lambda \) that makes the vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) coplanar, we can use the condition that the scalar triple product of the vectors must equal zero. This means that the dot product of one vector with the cross product of the other two must be zero.

Given:
- \( \vec{a} = \hat{i} + 3\hat{j} + 4\hat{k} \)
- \( \vec{b} = 2\hat{i} + \lambda\hat{j} + 2\hat{k} \)
- \( \vec{c} = 4\hat{i} - 7\hat{j} + 10\hat{k} \)

### Step 1: Calculate \( \vec{a} \times \vec{c} \)
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