# Linear Algebra 1 problem, Vector Geometry: Lines

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• Student323

#### Student323

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.

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

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.

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##.