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 is zero. The vectors are given as:

\[ \vec{a} = \hat{i} + 3\hat{j} + 4\hat{k} \]
\[ \vec{b} = 2\hat{i} + \lambda\hat{j} + 2\hat{k} \]
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