Parametric Equation for Line through (1,0,-1) and Parallel to 1/3(x-4)=1/2y=z+2

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In summary, to find the parametric equation for a line parallel to another line, we can use the direction vector of the original line and express the coordinates x, y, z in terms of the parameter t. In this case, the parametric equation for the line through (1,0,-1) and parallel to the line 1/3(x-4)=1/2y=z+2 is x=1+3t, y=2t, z=-1+t.
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TrueStar
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Homework Statement



Find the parametric equation for the line through (1,0,-1) and parallel to the line 1/3(x-4)=1/2y=z+2

Homework Equations



Vector equation - r=ro+tv

Perhaps the scaler equation? I'm not entirely sure.

The Attempt at a Solution



I'm not sure where to begin; I don't see any examples in the textbook to go on. I'm not sure if I can read the line given to me and pull out points to use and then create a vector with it.

Thanks
 
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  • #2
Find the direction vector of the original line from the equations

1/3 (x-4)=t, 1/2 y =t, z+2=t, expressing x, y, z with the parameter t.

ehild
 
  • #3
So when the line is written with the variables all equal to each other, I am to assume each one is actually equal to t?
 
  • #4
The above expressions of the coordinates x,y,z are all equal, so each of them is equal to the same number, call it t...

ehild
 
  • #5
Ah. I read through the text again and I think I understand how to read the equation better. I was looking for a set of points, but really I can get a vector out of it.

Anyway, I believe the answer is:
x= 1+3t
y= 2t
z= -1+t

Hopefully I am correct.
 
  • #6
Ah. I read through the text again and I think I understand how to read the equation better. I was looking for a set of points, but really I can get a vector out of it.

Anyway, I believe the answer is:
x= 1+3t
y= 2t
z= -1+t

Hopefully I am correct.
 
  • #7
TrueStar said:
Anyway, I believe the answer is:
x= 1+3t
y= 2t
z= -1+t

Hopefully I am correct.

You are! Good job!

ehild
 

FAQ: Parametric Equation for Line through (1,0,-1) and Parallel to 1/3(x-4)=1/2y=z+2

What is a parametric equation?

A parametric equation is a mathematical representation of a curve or surface in terms of one or more parameters. It allows for a more flexible and convenient way to describe a geometric figure or shape.

How do you find the parametric equation for a line?

To find the parametric equation for a line, you need to have the coordinates of at least two points on the line. Then, you can use the formula x = x0 + at and y = y0 + bt, where (x0,y0) is one of the given points, t is the parameter, and a and b are the slopes in the x and y directions, respectively.

What does it mean for a line to be parallel to a given line?

When a line is parallel to a given line, it means that the two lines never intersect and have the same slope. In other words, they have the same direction but different positions on the coordinate plane.

How do you determine the slope of a line from its equation?

The slope of a line can be determined from its equation by rearranging the equation into the slope-intercept form, y = mx + b, where m is the slope. Alternatively, you can use the formula m = (y2 - y1) / (x2 - x1), where (x1,y1) and (x2,y2) are two points on the line.

What are the steps to find the parametric equation for a line through a given point and parallel to a given line?

The steps to find the parametric equation for a line through a given point and parallel to a given line are:

  1. Find the slope of the given line using its equation.
  2. Use the slope and the given point to find the y-intercept of the new line.
  3. Write the equation of the new line in slope-intercept form, y = mx + b.
  4. Use the formula x = x0 + at and y = y0 + bt to write the parametric equations, where (x0,y0) is the given point and t is the parameter.

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