Thread: vector and parametric equations problem

Given two points P(1, 7) and Q(−2,−3) there exists a line which goes through these two points.
This line intersects the x-axis at the point x_0 = (x_0, 0).

Find the general, vector and parametric equations of a different line which goes through the point x_0 and is parallel to the vector v = (−1,−3).

Originally Posted by quah13579

Given two points P(1, 7) and Q(−2,−3) there exists a line which goes through these two points.
This line intersects the x-axis at the point x_0 = (x_0, 0).

Find the general, vector and parametric equations of a different line which goes through the point x_0 and is parallel to the vector v = (−1,−3).

The vector from P to Q is <1- (-2), 7- (-3)= <3, 10> so parametric equations for the line would be x= 3t+ 1, y= 10t+ 7. That intersects the x-axis when y= 10t+7= 0 or when t= -7/10. Then x= 3t+ 1= -21/10+ 1= -11/10. Parametric equations for the line through (-11/10, 0) parallel to <-1, -3> are x=-t+ 11/10, y= -3t. The vector equation would be [tex]\vec{r}(t)= (-t+ 11/10)\vec{i}- 3t/vec{j}. Solving each of the parametric equations for t, x= -t+ 11/10 gives t= -x+ 11/10 and y= -3t gives t= -y/3. Since those are both equal to t, they are equal to each other: -y/3= -x+ 11/10 or y= 3x- 33/10.