Question

Find a parametric representation for the line passing through p(1,3,-1) and q(-4,1,2).

Answer 1

To find the parametric representation of a line passing through two given points, subtract the coordinates of one point from the other to get the direction vector. Then, write the parametric equations using a parameter t and substitute the values to obtain the final representation.

Step-by-step explanation:

To find the parametric representation of a line passing through two given points, we can use the following steps:

1. Find the direction vector of the line by subtracting the coordinates of one point from the other. In this case, the direction vector is (1-(-4), 3-1, -1-2) = (5, 2, -3).
2. Choose any parameter t and write the parametric equations as x = x0 + t*dx, y = y0 + t*dy, and z = z0 + t*dz, where (x0, y0, z0) are the coordinates of one of the given points and (dx, dy, dz) are the components of the direction vector.
3. Substitute the values of the given point and the direction vector into the parametric equations to obtain the final parametric representation.

In this case, the parametric representation of the line passing through p(1,3,-1) and q(-4,1,2) is x = 1 + 5t, y = 3 + 2t, and z = -1 - 3t.