Finding Parametric Equations of a Line in Space

Find parametric equations of the line containing the point \((2,-3,1)\) and in the direction of the vector \(4\mathbf{i}+\dfrac{3}{4}\mathbf{j}-\mathbf{k}\).

Solution Let \((x_{0},y_{0},z_{0})=(2,-3,1)\) and \(a\mathbf{i}+b \mathbf{j}+c\mathbf{k}=4\mathbf{i}+\dfrac{3}{4}\mathbf{j}-\mathbf{k}\). Then, the parametric equations of the line are \[ x=x_{0}+at=2+4t\qquad y=y_{0}+bt=-3+\frac{3}{4}t\qquad z=z_{0}+ct=1-t \]