# Points on lines with parametric equations (linear algebra)

• fattycakez
In summary: What would a general point on the parametric lines look like? The x, y, z components give in the problem?( I haven't taken multi variable calc yet and this class assumes I will only use algebra to complete the homework)A completely different approach uses the hint above by taking the cross product of the two direction vectors. Then, take two arbitrary points, one from each line, and create a displacement vector between those two points. The shortest distance between the two lines will have to be a scalar multiple of the vector you obtained from the cross product.
fattycakez

## Homework Statement

"Let L1 be the line having parametric equations : x = 2 - s, y = -1 + 2s, z = 1+s and L2 be the line:
x = 1 +t, y = 2+ t, z =2t .

a. Do the lines intersect? If so, find the point of intersection.

b. Find the point P on the graph of L1 that is closest to the graph of L2 and find the point Q on the graph of L2 that is closest to the graph of L1. Hint: Use the fact that the vector PQ will be orthogonal to the direction vectors of both lines. "

## The Attempt at a Solution

In part a, I set the parametric equations equal to each other and solved for t and s. It looks like the lines do not intersect.

I'm not sure how to go about part b. How does the hint that the vector PQ will be orthogonal to the direction vectors help me?
The direction vectors would be:
L1 = (-1, 2, 1) L2 = (1, 1, 2)

Any help is greatly appreciated!

Last edited:
fattycakez said:

## Homework Statement

"Let L1 be the line having parametric equations : x = 2 - s, y = -1 + 2s, z = 1+s and L2 be the line:
x = 1 +2, y = 2+ t, z =2t .
Should the x-coordinate of L2 be x = 1 + 2t?
fattycakez said:
a. Do the lines intersect? If so, find the point of intersection.

b. Find the point P on the graph of L1 that is closest to the graph of L2 and find the point Q on the graph of L2 that is closest to the graph of L1. Hint: Use the fact that the vector PQ will be orthogonal to the direction vectors of both lines. "

## The Attempt at a Solution

In part a, I set the parametric equations equal to each other and solved for t and s. It looks like the lines do not intersect.

I'm not sure how to go about part b. How does the hint that the vector PQ will be orthogonal to the direction vectors help me?
The direction vectors would be:
L1 = (-1, 2, 1) L2 = (1, 1, 2)

Any help is greatly appreciated!

Ahh sorry, It should be x = 1+ t

fattycakez said:
I'm not sure how to go about part b. How does the hint that the vector PQ will be orthogonal to the direction vectors help me?
The direction vectors would be:
L1 = (-1, 2, 1) L2 = (1, 1, 2)
At the points on the lines that are closest to each other, the segment joining the two lines will be perpendicular to each line.

Okay, will it have something to do with PQ ⋅ L1 = 0 and PQ ⋅L2 = 0? Or am I way off here?

Mark's hint is different from the hint given in the original post. Take a general point on each line, one depends on ##s## and the other on ##t##. Write the distance (easier to use distance^2) between those two points. Minimize that using calculus methods to minimize a function of two variables. The ##s## and ##t## you get will give you the two points you seek.

What would a general point on the parametric lines look like? The x, y, z components give in the problem?
( I haven't taken multi variable calc yet and this class assumes I will only use algebra to complete the homework)

LCKurtz said:
Mark's hint is different from the hint given in the original post.
I don't see how what I said was different. In the OP, it states that "Hint: Use the fact that the vector PQ will be orthogonal to the direction vectors of both lines."
LCKurtz said:
Take a general point on each line, one depends on ##s## and the other on ##t##. Write the distance (easier to use distance^2) between those two points. Minimize that using calculus methods to minimize a function of two variables. The ##s## and ##t## you get will give you the two points you seek.

A completely different approach uses the hint above by taking the cross product of the two direction vectors. Then, take two arbitrary points, one from each line, and create a displacement vector between those two points. The shortest distance between the two lines will have to be a scalar multiple of the vector you obtained from the cross product.

## 1. What are parametric equations in linear algebra?

Parametric equations in linear algebra are a way to represent lines or curves in higher dimensions by using a set of variables, known as parameters. These parameters are typically represented by letters such as t or s, and they allow us to describe a point on the line or curve using a single equation.

## 2. How do you find points on a line with parametric equations?

To find points on a line with parametric equations, we need to have two equations, one for each coordinate (x and y). These equations will have the same parameter, and we can use it to solve for both coordinates simultaneously. The resulting values will be the x and y coordinates of the point on the line.

## 3. Can you use parametric equations to find the distance between two points on a line?

Yes, parametric equations can be used to find the distance between two points on a line. We can use the distance formula, which is based on the Pythagorean theorem, to calculate the distance between the two points using the x and y coordinates obtained from the parametric equations.

## 4. How do parametric equations relate to vectors?

Parametric equations and vectors are closely related in linear algebra. In fact, we can think of parametric equations as describing the path of a vector as it moves through different points on a line. The parameters in the equations represent the magnitude and direction of the vector.

## 5. What are some real-world applications of points on lines with parametric equations?

Parametric equations have many real-world applications, including in computer graphics, physics, and engineering. They are used to model and analyze motion, such as the trajectory of a projectile or the movement of a car along a curved path. They are also used in computer animation to create smooth and realistic movement. In engineering, parametric equations are used to design and analyze complex systems, such as the motion of robotic arms or the flow of fluids through pipes.

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