Undergrad Linear Algebra 1 problem, Vector Geometry: Lines

[Student323]

TL;DR

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.

 

[Office_Shredder]

Given a choice of t, can you write down the distance between L(t) and (-3,1)?

 

[fresh_42]

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.

 

[fresh_42]

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 $2$.