find a set of parametric equations for the line through the point and parallel to the specified vector.
(1, -8, -3), parallel to (-1, 6, -6)
find symmetric equations for the line through the point and parallel to the specified vector.
(5, -2, -4), parallel to (3, 6, 8)
1.
x = 1 - t
y = -8 + 6t
z = -3 - 6t
2.
(x-5)/3 = (y+2)/6 = (z+4)/8
did you notice the pattern in the two answers ?
Parametric equations for the line through (1, -8, -3), parallel to (-1, 6, -6) are:
x = 1 - t
y = -8 + 6t
z = -3 - 6t
Symmetric equations for the line through (5, -2, -4), parallel to (3, 6, 8) are:
(x - 5)/3 = (y + 2)/6 = (z + 4)/8
To find parametric equations for a line through a given point and parallel to a specified vector, we can use the following equations:
x = x0 + at
y = y0 + bt
z = z0 + ct
Where (x0, y0, z0) is the given point and (a, b, c) is the direction vector.
For the first question:
Given point: (1, -8, -3)
Direction vector: (-1, 6, -6)
Using the parametric equations, the set of parametric equations for the line can be written as:
x = 1 - t
y = -8 + 6t
z = -3 - 6t
For the second question:
Given point: (5, -2, -4)
Direction vector: (3, 6, 8)
Using the parametric equations, the set of parametric equations for the line can be written as:
x = 5 + 3t
y = -2 + 6t
z = -4 + 8t
To find the symmetric equations for a line through a given point and parallel to a specified vector, we can use the following equations:
(x - x0) / a = (y - y0) / b = (z - z0) / c
For the first question:
Given point: (1, -8, -3)
Direction vector: (-1, 6, -6)
Using the symmetric equations, the equations for the line can be written as:
(x - 1) / -1 = (y + 8) / 6 = (z + 3) / -6
For the second question:
Given point: (5, -2, -4)
Direction vector: (3, 6, 8)
Using the symmetric equations, the equations for the line can be written as:
(x - 5) / 3 = (y + 2) / 6 = (z + 4) / 8
To find a set of parametric equations for a line through a given point and parallel to a given vector, follow these steps:
1. Let the coordinates of the given point be (x₀, y₀, z₀).
2. Let the components of the given vector be (a, b, c).
3. Express the equations of the line in parametric form as:
x = x₀ + at
y = y₀ + bt
z = z₀ + ct
So, for the first problem, the set of parametric equations for the line through the point (1, -8, -3) parallel to the vector (-1, 6, -6) would be:
x = 1 - t
y = -8 + 6t
z = -3 - 6t
Now, let's move on to the second problem of finding symmetric equations for a line through a point and parallel to a vector:
1. Use the same given point, (x₀, y₀, z₀), and vector, (a, b, c), from the previous problem.
2. Express the equations of the line in symmetric form as:
(x - x₀) / a = (y - y₀) / b = (z - z₀) / c
So, for the second problem, the symmetric equations for the line through the point (5, -2, -4) and parallel to the vector (3, 6, 8) would be:
(x - 5) / 3 = (y + 2) / 6 = (z + 4) / 8