Find Parametric Equations for Perpendicular Line Through Point

In summary, the vector component V of a vector V from point P to point Q is the displacement vector V itselfminus the vector projection of V on the line from point P to Q.
  • #1
jcook735
33
0
Find parametric equations for the line through the point that is perpendicular to the given line, and intersects this line.

We are given the point (0, 2, 2) and the line
x = 1 + t, y = 2 - t, z = 2t.

We are also given one of the parametric equations, that being x=3s. I am looking for the y and z parametric equations.


I figured that the equation for the first line is = (1, 2, 0) + t(1, -1, 2)
and the equation for the second line is (0,2,2) + s(3,b,c).



Beyond that, I have no idea what to do.
 
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  • #2


jcook735 said:
Find parametric equations for the line through the point that is perpendicular to the given line, and intersects this line.

We are given the point (0, 2, 2) and the line
x = 1 + t, y = 2 - t, z = 2t.

We are also given one of the parametric equations, that being x=3s. I am looking for the y and z parametric equations.


I figured that the equation for the first line is = (1, 2, 0) + t(1, -1, 2)
and the equation for the second line is (0,2,2) + s(3,b,c).



Beyond that, I have no idea what to do.

Have you had how to calculate the vector component of one vector on another? If so, here's a hint:

Consider a point P on the given line, for example P = (1,2,0). Consider the vector V from that point to your point (0,2,2). Calculate the vector component of V on your line. You can use that and your point P to calculate the point Q where the perpendicular line would intersect. Knowing P and Q the rest is easy...
 
  • #3


I feel like I am missing something really obvious, but I don't understand how i could use point P and the vector component V to find point Q. The vector component V is (-1,0,2), correct?
 
  • #4


LCKurtz said:
Have you had how to calculate the vector component of one vector on another? If so, here's a hint:

Consider a point P on the given line, for example P = (1,2,0). Consider the vector V from that point to your point (0,2,2). Calculate the vector component of V on your line. You can use that and your point P to calculate the point Q where the perpendicular line would intersect. Knowing P and Q the rest is easy...

jcook735 said:
I feel like I am missing something really obvious, but I don't understand how i could use point P and the vector component V to find point Q. The vector component V is (-1,0,2), correct?

"Vector component V" doesn't make sense.

It is the displacement vector V itself that is <-1,0,2>. That isn't the same thing as the vector projection of V on your line.

Call your point (0,2,2) S. Draw a picture of a line noting the point P(1,2,0) on the line and the point S(0,2,2) off the line and the displacement vector V going from P to S. If you drop a perpendicular from S to the line, hitting it at Q, you get a little right triangle. Note that no matter what point on the line you use for P, you always get the same Q.

Also notice that if you add the components of the vector PQ to the coordinates of P you will get the coordinates of Q. And once you have Q you can write the equation of the line you seek.

The vector PQ is called the vector projection of V on the line. It is what you need to calculate to solve your problem, which is why I asked if you had studied vector projections yet. Does that help?
 
  • #5


Ohhhhh yes I understand now. Thank you!
 

What are parametric equations?

Parametric equations are a set of equations that express the coordinates of a point on a curve or surface as functions of one or more independent variables, called parameters.

How can I find the parametric equations for a perpendicular line through a point?

To find the parametric equations for a perpendicular line through a point, you will need the coordinates of the point and the slope of the line. You can then use the point-slope form of a line to determine the equations.

What is the point-slope form of a line?

The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is the given point on the line and m is the slope.

Can parametric equations be used to represent any type of curve or surface?

Yes, parametric equations can be used to represent any type of curve or surface, including lines, circles, parabolas, ellipses, and more complex curves and surfaces.

Are there any advantages to using parametric equations over other forms of equations?

Parametric equations have several advantages over other forms of equations, including their ability to easily represent curves and surfaces in higher dimensions and their usefulness in applications requiring motion or variation, such as physics and engineering problems.

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