Represent the line segment from P to Q by a vector-valued function. (P corresponds to t = 0. Q corresponds to t = 1.)

Question

P(−6, −1, −1), Q(−2, −8, −4)

r(t) = ____

Represent the line segment from P to Q by a set of parametric equations. (Enter your answers as a comma-separated list of equations.)

Answer 1

The first step to solving this problem is to calculate the direction vector from P to Q and use it with point P to develop the vector-valued function.

The direction vector PQ can be found by subtracting the coordinates of P from the coordinates of Q:

PQ = Q - P

PQ = <-2 - (-6), -8 - (-1), -4 - (-1)>

PQ = <4, -7, -3>

The formula for a line segment connecting a point P to a point Q is given by r(t) = P + tPQ. Substituting the coordinates of P and the components of the vector PQ, we get

r(t) = <-6, -1, -1> + t<4, -7, -3> = <-6 + 4t, -1 - 7t, -1 - 3t>.

In parametric form, the list of comma-separated equations is given by:

x = -6 + 4t, y = -1 - 7t, z = -1 - 3t.

Hope this helped!