Question

Determine parametric equations of the line segment from point p(4, 5, −1) to q(−7, 2, 3). (enter your answers as a comma-separated list of equations. Let 0 ≤ t ≤ 1 be the parameter.)

Answer 1

we are given two points as

p=(4, 5, −1)

q=(−7, 2, 3)

we have to find parametric equation

firstly, we will find pq

$pq=q-p$

$pq=(-7,2,3)-(4,5,-1)$

$pq=(-7-4,2-5,3+1)$

$pq=(-11,-3,4)$

now, we can find equation of line

L: p +t(pq)

so, we get

$L=(4,5,-1)+t(-11,-3,4)$

$(x,y,z)=(4-11t,5-3t,-1+4t)$

$x(t)=4-11t$

$y(t)=5-3t$

$z(t)=-1+4t$................Answer

Answer 2

We will see that the parametric equation that connects the points is:

(4 - 11t, 5 - 3t, -1 + 4t)

for 0 ≤ t ≤ 1

Finding the equation for the segment.

The general parametric equation that connects the points (x, y, z) to (x', y', z') is given by:

(1 - t)*(x, y, z) + t*(x', y', z')

for 0 ≤ t ≤ 1

You can see that when t = 0, we start at (x, y, z), and when t = 1 we end at (x', y', z').

Now we replace our points (4, 5, -1) and (-7, 2, 3) so we get:

(1 - t)*(4, 5, -1) + t*(-7, 2, 3)

= (4 - 4t - 7t, 5 - 5t + 2t,  -1 + t + 3t)

= (4 - 11t, 5 - 3t, -1 + 4t)

Then the parametric equation that represents the segment is:

(4 - 11t, 5 - 3t, -1 + 4t)

for 0 ≤ t ≤ 1

If you want to learn more about parametric equations, you can read:

https://brainly.com/question/12695467