Finding if a point belongs to a line in space.

[smithnya]

Hello everyone,

I have a simple question, but I am unsure. I know from a point p₀ = (x₁, y₁) and a vector v = <a, b>, I can obtain a parametrized set of equations for a line in space such that x = x₁ + at and y = y₁ + bt. How can I check that any other point, not p₀, is/isn't in that line?

 

[micromass]

For example, let's say you have x=2+3t, y=3+4t

Then a number (x,y) lies on the lies if and only if there exist a t such that x=2+3t AND y=3+4t

Let's see if (0,0) is on the line. If (0,0) were on the line, then there would exist a t such that 0=2+3t AND 0=3+4t. Solving the equations gets us that t=-2/3 AND t=-3/4. This is clearly false (t can not be two values at once). Thus (0,0) is not on the line.

Take (8,11). If this were on the line, then there would exist a t such that 8=2+3t AND 11=3+4t. Solving the equations gets us t=2 AND t=2. So such a t exists (and equal 2). Thus (8,11) is on the line.

 

[smithnya]

Thank you so much. That was very simple and it explained what I needed to know. Nice avatar by the way. I am listening to "The Great Gig in the Sky" as I type.

 

[micromass]

Thank you so much. That was very simple and it explained what I needed to know. Nice avatar by the way. I am listening to "The Great Gig in the Sky" as I type.

You have a great taste in music!