Find parametric equations of lines through
(-5, -6, 8) and (1, 3, 7) b. (10, 3, 1) and (6, -2, -3)
To find parametric equations of a line, we need a direction vector and a point on the line.
For the line through (-5, -6, 8) and (1, 3, 7):
1. Find the direction vector:
Let v be the vector from (-5, -6, 8) to (1, 3, 7).
v = (1 - (-5), 3 - (-6), 7 - 8)
= (6, 9, -1)
2. Choose a point on the line, say (-5, -6, 8). Let t be a parameter.
Therefore, the parametric equations for this line are:
x = -5 + 6t
y = -6 + 9t
z = 8 - t
For the line through (10, 3, 1) and (6, -2, -3):
1. Find the direction vector:
Let w be the vector from (10, 3, 1) to (6, -2, -3).
w = (6 - 10, -2 - 3, -3 - 1)
= (-4, -5, -4)
2. Choose a point on the line, say (10, 3, 1). Let s be a parameter.
Therefore, the parametric equations for this line are:
x = 10 - 4s
y = 3 - 5s
z = 1 - 4s
To find the parametric equations of a line, we need to determine the direction vector of the line and a point on the line.
Let's start with the first set of points:
Point A: (-5, -6, 8)
Point B: (1, 3, 7)
1. Determine the direction vector:
The direction vector can be found by subtracting the coordinates of Point A from Point B:
Direction vector = (1 - (-5), 3 - (-6), 7 - 8)
= (6, 9, -1)
2. Choose a parameter, let's call it t.
3. Determine the parametric equations:
x = x0 + t * direction vector x-component
y = y0 + t * direction vector y-component
z = z0 + t * direction vector z-component
Where (x0, y0, z0) is a point on the line.
Using Point A(-5, -6, 8) as a point on the line, the parametric equations are:
x = -5 + 6t
y = -6 + 9t
z = 8 - t
Now let's move on to the second set of points:
Point C: (10, 3, 1)
Point D: (6, -2, -3)
1. Determine the direction vector:
The direction vector can be found by subtracting the coordinates of Point C from Point D:
Direction vector = (6 - 10, -2 - 3, -3 - 1)
= (-4, -5, -4)
2. Choose a parameter, let's call it t.
3. Determine the parametric equations:
x = x0 + t * direction vector x-component
y = y0 + t * direction vector y-component
z = z0 + t * direction vector z-component
Where (x0, y0, z0) is a point on the line.
Using Point C(10, 3, 1) as a point on the line, the parametric equations are:
x = 10 - 4t
y = 3 - 5t
z = 1 - 4t
So the parametric equations for the line through (-5, -6, 8) and (1, 3, 7) are:
x = -5 + 6t
y = -6 + 9t
z = 8 - t
And the parametric equations for the line through (10, 3, 1) and (6, -2, -3) are:
x = 10 - 4t
y = 3 - 5t
z = 1 - 4t