Find the parametric equations of the line through the points (2, -1, 3) and (-1, 4, 0).
get a direction vector:
v = [2+1, -1-4 , 3-0] = [ 3, -5, 3]
pick one of the points, let's pick (2,-1,3)
x = 2 + 3t
y = -1 - t
z = 3 + 3t
all done
Thank you so much!!
To find the parametric equations of a line passing through two points, we need to determine the position vector of the line in terms of a parameter t.
Let the line pass through point A(2, -1, 3) and B(-1, 4, 0).
We can start by finding the direction vector of the line which is represented by the vector AB.
The direction vector can be calculated by subtracting the coordinates of point A from those of point B:
AB = <(-1 - 2), (4 - (-1)), (0 - 3)>
= <-3, 5, -3>
Therefore, the direction vector of the line is < -3, 5, -3 >.
Now, we can use the general form of the parametric equations of a line in terms of a parameter t:
x = x0 + at
y = y0 + bt
z = z0 + ct
where (x0, y0, z0) is the starting point and (a, b, c) are the direction ratios.
Using point A(2, -1, 3) as the starting point and the direction vector <-3, 5, -3> as the direction ratios, the parametric equations of the line are:
x = 2 + (-3)t
y = -1 + 5t
z = 3 + (-3)t
Therefore, the parametric equations of the line through the points (2, -1, 3) and (-1, 4, 0) are:
x = 2 - 3t
y = -1 + 5t
z = 3 - 3t