What would the vector, parametric, and if possible, symmetric equations of the line through C(2, -2, 1) and parallel to the line with parametric equations x = -1 + 5t, y = 2 – t, z = 3 – 4t.

you have ignored the new point and are simply using the original given equation

Since the new line is parallel to the old one, simply edit your equations using the
new point.
Your given equation goes through the point (-1, 2, 3) , replace it with (2, -2, 1)

I also noted that in your answer for the symmetric equation you had the

direction vector incorrect.

Using the new point, it should have been
(x - 2)/5 = (y + 2)/-1 = (z - 1)/-4

Here's my attempt.

Parametric equation: x = -1 + 5t, y = 2 - t, z = 3 - 4t since the lines are parallel.
Vector equation: (-1, 2, 3) + t(5, -1, -4)
Symmetric equation: (x + 1)/5 = (y - 2)/1 = (z - 3)/4

Can someone double check this for me please?

Okay so the parametric eq'n would be: x = 2 + 5t, y = -2 - t, z = 1 - 4t and the vector would then be: (2, -2, -4) + t (5, -1, -4)

Made a mistake w/ the vector one. It should be (2, -2, 1)

To find the vector, parametric, and symmetric equations of a line through a point and parallel to another line, you can follow these steps:

Step 1: Find the direction vector of the given line.
The direction vector of the given line is obtained by taking the coefficients of t in the parametric equations. In this case, the direction vector is:
d = <5, -1, -4>

Step 2: Set up the vector equation of the line.
The vector equation of a line is given by: r = r0 + t * d, where r is a position vector, r0 is a specific point on the line, d is the direction vector, and t is a parameter.

Since we are given the point C(2, -2, 1), r0 = <2, -2, 1>.

The vector equation of the line is: r = <2, -2, 1> + t * <5, -1, -4>

Step 3: Convert the vector equation into parametric equations.
To obtain the parametric equations, we need to separate x, y, and z components.

x = x0 + at, y = y0 + bt, z = z0 + ct, where a, b, c are the components of the direction vector.

For this line:
x = 2 + 5t
y = -2 - t
z = 1 - 4t

These are the parametric equations of the line.

Step 4: Convert the parametric equations into symmetric equations. (Optional)
The symmetric equations represent the line in terms of equations involving x, y, and z, rather than parameter t.

To obtain the symmetric equations, we equate the parameter t to a variable, say s.

x - 2 = 5t
t = (x - 2)/5

By substituting this value of t into the y and z equations, we get the symmetric equations:

y = -2 - ((x - 2)/5)
z = 1 - 4((x - 2)/5)

These are the symmetric equations of the line.

To summarize:
Vector equation: r = <2, -2, 1> + t * <5, -1, -4>
Parametric equations: x = 2 + 5t, y = -2 - t, z = 1 - 4t
Symmetric equations: y = -2 - ((x - 2)/5), z = 1 - 4((x - 2)/5)