Linear Algebra 1 problem, Vector Geometry: Lines

In summary: The line can be described by ##x=-3+t## and ##y=1-2t##. The length between ##\binom{-3}{1}## and ##\binom{-3}{1}+t\cdot \binom{1}{-2}## is calculated by the distance formula.In summary, to find all x on the line L that lie 2 units from (-3, 1), you can either intersect the line with a circle around (-3,1) with radius 2 or calculate the length between the two points using the distance formula. The line can be described by x = -3 + t and y = 1 - 2t.
  • #1
Student323
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TL;DR Summary
Given the line L: x = (-3, 1) + t(1,-2) find all x on L that lie 2 units from (-3, 1).
Problem: Given the line L: x = (-3, 1) + t(1,-2) find all x on L that lie 2 units from (-3, 1).

I know the answer is (3 ± 2 / √5, -1 ± 4/√5) but I don't know where to start. I found that if t=2, x= (-5, 5) and the normal vector is (2, 1) but I am not sure if this information is useful or how to use it.
 
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  • #2
Given a choice of t, can you write down the distance between L(t) and (-3,1)?
 
  • #3
Student323 said:
Summary:: Given the line L: x = (-3, 1) + t(1,-2) find all x on L that lie 2 units from (-3, 1).

Problem: Given the line L: x = (-3, 1) + t(1,-2) find all x on L that lie 2 units from (-3, 1).

I know the answer is (3 ± 2 / √5, -1 ± 4/√5) but I don't know where to start. I found that if t=2, x= (-5, 5) and the normal vector is (2, 1) but I am not sure if this information is useful or how to use it.
You could intersect the line with a circle around (-3,1) with radius 2. For that set up the equation for the circle and use it for (x,y) on the line.
 
  • #4
Another way is to step along the line until you get to a point that is ##2## units apart. Calculate the length between ##\binom{-3}{1}## and ##\binom{-3}{1}+t\cdot \binom{1}{-2}## which equals ##2##.
 

Related to Linear Algebra 1 problem, Vector Geometry: Lines

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 vectors, matrices, and linear transformations.

2. What is a vector in Linear Algebra?

A vector in Linear Algebra is a mathematical object that has both magnitude and direction. It is represented as an ordered list of numbers and can be used to represent physical quantities such as velocity, force, and displacement.

3. How do you solve a Linear Algebra 1 problem?

To solve a Linear Algebra 1 problem, you first need to understand the concepts of vectors, matrices, and linear transformations. Then, you can use various techniques such as Gaussian elimination, matrix operations, and vector projections to solve the problem.

4. What is Vector Geometry?

Vector Geometry is the study of geometric objects using vectors. It involves the use of vectors to represent points, lines, and planes in space. It is an important concept in Linear Algebra as it helps in solving problems related to lines and planes.

5. How is Vector Geometry used to solve problems related to lines?

Vector Geometry is used to solve problems related to lines by representing the line as a vector equation and using techniques such as dot product, cross product, and parametric equations to find the properties of the line, such as its direction, slope, and distance from a point.

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