Finding Symmetric Equations of a Line in Space

Find symmetric equations of the line containing the point \((1,-1,2)\) and in the direction of the vector \(5\mathbf{i}-2\mathbf{j}+3\mathbf{k}\).

Solution The components of the vector \(5\mathbf{i}-2\mathbf{j}+3 \mathbf{k}\) are all nonzero. So, we use \(a=5\), \(b=-2\), and \(c=3\) and the coordinates of the point \((1,-1,2) \) to obtain the symmetric equations \[ \frac{x-1}{5}=\frac{y+1}{-2}=\frac{z-2}{3} \]