Find parametric equations for the line that passes through the point (0, -1, 1) and the midpoint of the line segment from (2, 3, -2) to (4, -1, 5).

Question

Find parametric equations for the line that passes through the point (0, -1, 1) and the midpoint of the line segment from (2, 3, -2) to (4, -1, 5).

I don't know how to solve this problem.

Answer 1

let M be the midpoint

then M =(3,1,3/2)

then a direction vector of the line is
[3-0,1+1,3/2-1] = [3,2,1/2] or [6,4,1}

possible parametric equations:

x = 0 + 6t
y = -1 + 4t
z = 1 + t

Answer 2

Well, you're in luck because I happen to be an expert in solving problems and making people laugh! So, let's tackle this math problem with a smile, shall we?

First, let's find the midpoint of the line segment. The midpoint is simply the average of the coordinates of the two endpoints.

Midpoint = ((2 + 4) / 2, (3 + (-1)) / 2, (-2 + 5) / 2)
= (3, 1, 1.5)

Great! Now we have the midpoint of the line segment. To find the parametric equations, we need a direction vector for the line. We can easily find this by subtracting the coordinates of the two points.

Direction vector = (4 - 2, -1 - 3, 5 - (-2))
= (2, -4, 7)

We have the midpoint and the direction vector. Now, we can write the parametric equations for the line. Let's call the parameter t.

x(t) = 3 + 2t
y(t) = 1 - 4t
z(t) = 1.5 + 7t

So, the parametric equations for the line passing through the point (0, -1, 1) and the midpoint of the line segment from (2, 3, -2) to (4, -1, 5) are:

x(t) = 3 + 2t
y(t) = 1 - 4t
z(t) = 1.5 + 7t

I hope my humor made solving this problem a little more enjoyable for you! If you have any more questions, feel free to ask!

Answer 3

To find the parametric equations for the line passing through a point and the midpoint of the line segment, you can follow these steps:

Step 1: Find the midpoint of the line segment.
Given two points (2, 3, -2) and (4, -1, 5), the midpoint can be found by taking the average of the coordinates:
Midpoint = [(2 + 4) / 2, (3 + (-1)) / 2, (-2 + 5) / 2] = (3, 1, 1.5)

Step 2: Find the direction vector of the line segment.
The direction vector of the line segment can be found by subtracting the coordinates of the two points:
Direction Vector = (4, -1, 5) - (2, 3, -2) = (2, -4, 7)

Step 3: Determine the parametric equations.
The parametric equations for the line passing through the point (0, -1, 1) and the midpoint (3, 1, 1.5) can be written as follows:
x = 0 + 2t
y = -1 - 4t
z = 1 + 7t

Here, t is the parameter that represents any point on the line. By substituting different values of t, you can obtain the corresponding coordinates on the line.

Answer 4

To find the parametric equations for the line, we need to determine the direction vector and a point on the line.

First, let's find the midpoint of the line segment from (2, 3, -2) to (4, -1, 5). The midpoint formula is given by:

Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)

Plugging in the coordinates, we have:

Midpoint = ((2 + 4)/2, (3 + -1)/2, (-2 + 5)/2)
= (6/2, 2/2, 3/2)
= (3, 1, 3/2)

Now, we have the point (3, 1, 3/2) on the line.

To find the direction vector, we subtract the coordinates of the given point (0, -1, 1) from the coordinates of the midpoint:

Direction vector = (3, 1, 3/2) - (0, -1, 1)
= (3, 1 + 1, 3/2 - 1)
= (3, 2, 1/2)

Now, we can write the parametric equations for the line:

x = 3t
y = 1 + 2t
z = 3/2 + (1/2)t

where t is a parameter.

So, the parametric equations for the line passing through the point (0, -1, 1) and the midpoint of the line segment from (2, 3, -2) to (4, -1, 5) are:
x = 3t
y = 1 + 2t
z = 3/2 + (1/2)t