Find the parametric equations of the line through the points (1, 3, 2) and (4, -1, 1).
The direction numbers of the line joining the two points are (3,-4,-1) so
x = 1+3t
y = 3-4t
z = 2-t
or, eliminating t,
(x-1)/3 = (y-4)/-4 = (z-2)/-1
To find the parametric equations of the line through two points, we need to find a vector that lies on the line and an initial point on the line.
Step 1: Find the direction vector of the line
To find the direction vector, we subtract the coordinates of one point from the other point. Let's subtract (1, 3, 2) from (4, -1, 1):
Direction vector = (4, -1, 1) - (1, 3, 2)
= (3, -4, -1)
Step 2: Find an initial point on the line
We can take either of the given points as an initial point. Let's choose (1, 3, 2).
Step 3: Write the parametric equations
The parametric equations of a line in three-dimensional space are given by:
x = x₀ + at
y = y₀ + bt
z = z₀ + ct
where (x, y, z) is a point on the line, (x₀, y₀, z₀) is an initial point on the line, and (a, b, c) is the direction vector.
Plugging in the values for the initial point and the direction vector, we get:
x = 1 + 3t
y = 3 - 4t
z = 2 - t
Therefore, the parametric equations of the line passing through the points (1, 3, 2) and (4, -1, 1) are:
x = 1 + 3t
y = 3 - 4t
z = 2 - t
To find the parametric equations of the line passing through two points, we need to find the direction vector of the line and the coordinates of one point on the line.
Let's start by finding the direction vector of the line. We can obtain the direction vector by subtracting the coordinates of one point from the other point. Using the points (1, 3, 2) and (4, -1, 1), we have:
Direction vector = (4 - 1, -1 - 3, 1 - 2)
= (3, -4, -1)
Next, we need to find the coordinates of one point on the line. Let's take the coordinates of the first point, (1, 3, 2).
Now, we can write the parametric equations of the line using the coordinates of the point on the line and the direction vector:
x = 1 + 3t
y = 3 - 4t
z = 2 - t
Therefore, the parametric equations of the line passing through the points (1, 3, 2) and (4, -1, 1) are:
x = 1 + 3t
y = 3 - 4t
z = 2 - t