Find the vector equation of the line passing through the point \((\hat i+2\hat j+3\hat k)\) and perpendicular to the vectors \(\hat i+\hat j+\hat k\) and \(2\hat i-\hat j+\hat k\).

Answer: \(\bar r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+\hat j-3\hat k)\)

A vector perpendicular to both \(\bar b_1=\hat i+\hat j+\hat k\) and \(\bar b_2=2\hat i-\hat j+\hat k\) is

\(\bar b=\bar b_1\times\bar b_2=\begin{vmatrix}\hat i&\hat j&\hat k\\1&1&1\\2&-1&1\end{vmatrix}=\hat i(1+1)-\hat j(1-2)+\hat k(-1-2)=2\hat i+\hat j-3\hat k\).

The line through \(\bar a=\hat i+2\hat j+3\hat k\) parallel to \(\bar b\) is \(\bar r=\bar a+\lambda\bar b\):

\(\bar r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+\hat j-3\hat k)\), \(\lambda\) a scalar.