# [Linear Algebra] - Find the shortest distance d between two lines

• MHB
• CoolMan2017
In summary: Once you have s and t, use those equations to find Q1 and Q2. Since you have not shown any work, I cannot know where you need help. Please show us your work so far and explain what you don't understand.
CoolMan2017
Let L1 be the line passing through the point P1=(−2,−11,9) with direction vector d2=[0,2,−2]T, and let L2 be the line passing through the point P2=(−2,−1,11) with direction vector d2=[−1,0,−1]T Find the shortest distance d between these two lines, and find a point Q1 on L1 and a point Q2 on L2 so that d(Q1,Q2)=d. Use the square root symbol to get the exact value

Hello and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

Looks pretty straight forward to me- though, frankly, I don't see what it has to do with "Linear Algebra"- just basic Calculus.

Line L1 can be written in a vector equation as (x, y, z)= (-2, -11, 9)+ (0, 2, -2)t= (-2, -11+ 2t, 9- 2t) and L2 as (x, y, z)= (-2, -1, 11)+ (-1, 0, -1)s= (-2- s, -1, 11- s). The distance between any two points, one on the first line, the other on the second is $D= \sqrt{(-2- (-2-s))^2+ (-11+ 2t- (-1))^2+ (9- 2t- (11- s)^2}= \sqrt{s^2+ (-10+ 2t)^2+ (s- 2t- 2)^2}$.

Minimizing that distance is the same as minimizing its square:
$D^2= s^2+ (-10+ 2t)^2+ (s- 2t- 2)^2$.

Set the derivatives with respect to s and t to 0 and solve the two equations for s and t.

## 1. What is Linear Algebra?

Linear Algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It involves the study of operations on vectors and matrices and how these can be used to solve systems of linear equations.

## 2. What is the shortest distance between two lines?

The shortest distance between two lines is the length of the perpendicular line segment that connects the two lines. This distance can be calculated using the formula d = |(P1-P2)·n|/|n|, where P1 and P2 are points on the two lines and n is the normal vector to both lines.

## 3. How can Linear Algebra be used to find the shortest distance between two lines?

Linear Algebra provides the tools and techniques to represent and manipulate linear equations and systems. By representing the two lines as equations in vector form, we can use the formula mentioned above to calculate the shortest distance between them.

## 4. Are there any practical applications of finding the shortest distance between two lines?

Finding the shortest distance between two lines is useful in various fields such as computer graphics, robotics, and engineering. It can be used to determine the closest distance between two objects or to calculate the shortest path for a robot to move from one point to another.

## 5. Is there a difference between the shortest distance between two lines and the distance between two parallel lines?

Yes, there is a difference between the shortest distance between two lines and the distance between two parallel lines. The shortest distance between two lines is the length of the perpendicular line segment that connects the two lines, while the distance between two parallel lines is the distance between any two points on the two parallel lines.

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