# Find parametric equations for the line through the points P=(-4,5,6) and Q=(3,-2,4)

## Question:

Find parametric equations for the line through the points {eq}P=(-4,5,6) {/eq} and {eq}Q=(3,-2,4) {/eq}

## Parametric Equation of the Line Segment:

A solution to find the parametric equation of a line segment is to first define its direction vector and use a point that passes through the line. If we have two points that are on the line we can establish their vector direction whose components are the difference in coordinates of the two given points. We then substitute these values of the direction vector and the point in the formula of the parametric equation of the line.

{eq}\eqalign{ & {\text{The parametric equation of a line segment containing the point }}P\left( {{x_1},{y_1},{z_1}} \right){\text{, }}Q\left( {{x_2},{y_2},{z_2}} \right){\text{ and }} \cr & {\text{parallel to a vector }}\,\vec v = \left\langle {{v_1},{v_2}} \right\rangle \,{\text{ is given by:}} \cr & \,\,\,\,\,\,x = {x_0} + {v_1}t,\,\,\,y = {y_0} + {v_2}t,\,\,\,z = {z_0} + {v_3}t \cr & {\text{where }}\vec v = \overrightarrow {PQ} = \left\langle {{v_1},{v_2},{v_3}} \right\rangle = \left\langle {{x_2} - {x_1},{y_2} - {y_1},{z_2} - {z_3}} \right\rangle \cr & {\text{In this case }}\,{\text{we have that the segment line contains the points }}\,P\left( { - 4,5,6} \right){\text{ and }}\,Q\left( {3, - 2,4} \right){\text{, then }} \cr & {\text{its direction vector is determined by:}} \cr & \,\,\,\vec v = \overrightarrow {PQ} = \left\langle {3 + 4, - 2 - 5,4 - 6} \right\rangle = \left\langle {7, - 7, - 2} \right\rangle \cr & {\text{Then}}{\text{, substituting }}\,P\left( {{x_1},{y_1},{z_1}} \right) = P\left( { - 4,5,6} \right)\,{\text{ and }}\,\vec v = \left\langle {{v_1},{v_2},{v_3}} \right\rangle = \left\langle {7, - 7, - 2} \right\rangle \,{\text{ in the parametric }} \cr & {\text{equation of the line:}} \cr & \,\,\,\,\,x = {x_0} + {v_1}t,\,\,\,y = {y_0} + {v_2}t,\,\,\,z = {z_0} + {v_3}t \cr & \,\,\,\,\,x = - 4 + 7t,\,\,\,y = 5 - 7t,\,\,\,z = 6 - 2t \cr & {\text{Therefore}}{\text{, the parametric equation of the line is:}} \cr & \,\,\,\,\,\boxed{x = - 4 + 7t,\,\,\,y = 5 - 7t,\,\,\,z = 6 - 2t} \cr} {/eq}