Solving Parametric Equations for Line Through Point & Parallel to Given Line

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Homework Statement

Find a vector equation and parametric equations in t for the line through the point and parallel to the given line.
(0, 12, -11)
x = -5 + 3t, y = 4 - 2t, z = 1 + 8t

Homework Equations

x = x0 + at y = y0 + bt z = z0 + ct

The Attempt at a Solution

I know how to work this problem if I'm given the point and the vector,


when you have a point(a,b,c) and vector (x,y,z) you just do

r = (ai + bj + ck) + t(xi + yj + zk)


...but I can't figure out how to do it when the line is given like this.

 

[mighty2000]

To find the vector equation and parametric equations for a line through a given point and parallel to a given line, you can use the following steps:

1. First, we need to find the direction vector of the given line. This can be done by looking at the coefficients of t in the equations for x, y, and z. In this case, the direction vector is (3, -2, 8).

2. Next, we can use the point-slope form of a line to find the vector equation. This form is given by r = r0 + t*v, where r0 is the given point and v is the direction vector. In this case, the vector equation is:

r = (0, 12, -11) + t*(3, -2, 8)

3. To find the parametric equations, we can simply break down the vector equation into its components. This gives us:

x = 0 + 3t = 3t
y = 12 - 2t
z = -11 + 8t

Therefore, the parametric equations for the line through the point (0, 12, -11) and parallel to the given line are:

x = 3t
y = 12 - 2t
z = -11 + 8t

I hope this helps. Let me know if you have any further questions.