Find parametric equations of lines through
a. (-5, -6, 8) and (1, 3, 7) b. (10, 3, 1) and (6, -2, -3)
a. To find the parametric equations of the line through (-5, -6, 8) and (1, 3, 7), we can use the following formula:
x = x1 + at
y = y1 + bt
z = z1 + ct
where (x1, y1, z1) is a point on the line, and (a, b, c) is the direction vector of the line.
First, we need to find the direction vector of the line. We can simply subtract the coordinates of the two points to get the direction vector:
a = 1 - (-5) = 6
b = 3 - (-6) = 9
c = 7 - 8 = -1
Now we can use the formula to write the parametric equations:
x = -5 + 6t
y = -6 + 9t
z = 8 - t
b. To find the parametric equations of the line through (10, 3, 1) and (6, -2, -3), we can again use the formula:
x = x1 + at
y = y1 + bt
z = z1 + ct
where (x1, y1, z1) is a point on the line, and (a, b, c) is the direction vector of the line.
First, we need to find the direction vector of the line:
a = 6 - 10 = -4
b = -2 - 3 = -5
c = -3 - 1 = -4
The direction vector is (-4, -5, -4).
Now we can use the formula to write the parametric equations:
x = 10 - 4t
y = 3 - 5t
z = 1 - 4t
a. To find the parametric equations of a line passing through the points (-5, -6, 8) and (1, 3, 7), we can use the following equation:
r = a + t * (b - a)
where r is the position vector of a point on the line, a is the coordinates of the first point, b is the coordinates of the second point, and t is a parameter.
Let's substitute the values into the equation:
r = (-5, -6, 8) + t * ((1, 3, 7) - (-5, -6, 8))
= (-5, -6, 8) + t * (6, 9, -1)
So, the parametric equations of the line passing through (-5, -6, 8) and (1, 3, 7) are:
x = -5 + 6t
y = -6 + 9t
z = 8 - t
b. Similarly, for the line passing through (10, 3, 1) and (6, -2, -3), we can use the same equation:
r = a + t * (b - a)
Substitute the values into the equation:
r = (10, 3, 1) + t * ((6, -2, -3) - (10, 3, 1))
= (10, 3, 1) + t * (-4, -5, -4)
So, the parametric equations of the line passing through (10, 3, 1) and (6, -2, -3) are:
x = 10 - 4t
y = 3 - 5t
z = 1 - 4t