How to get every x and y positions of 2 points

Question

I don't know if this question is already asked but can't find a solution for it.

Example i have these 2 points on the x-y plane $(2,1)$ and $(5,4)$

And then connects it to make a line

How am i able to get a formula for the x and y positions of other points on the line?

Answer 1

In order to find a line that passes through any two points $(x_0,y_0)$ and $(x_1,y_1)$ you must first find the slope, which is given by $\frac{y_1-y_0}{x_1-x_0}$. Then you must find the y-intercept, which is the difference between $y_0$ and the value of $\frac{y_1-y_0}{x_1-x_0}x_0$. In the end, your line will be given by the formula $$y=\frac{y_1-y_0}{x_1-x_0}x+y_0-\frac{y_1-y_0}{x_1-x_0}x_0$$ To find the coordinates of other points on the line, plug in the x-values of the points into this formula.

Answer 2

Say you have two points $A $ and $B $.

$B-A $ can be thought of as an arrow pointing from $A $ to $B $. Knowing that, take the expression

$$P_t = A + t (B-A) $$

where $0 \leq t \leq 1$.

Every such $P_t $ is in the line between $A $ and $B $.

If you write $A $ and $B $ in terms of its coordinates, you get what you want.

Answer 3

You have the points $\vec a=(2,1)$ and $\vec b=(5,4)$, so the vector between the two points is given by $\vec b- \vec a=(3,3)$. The points on the line are then given by the equation $\vec x=(2,1)+t(3,3)$, with $0\leq t\leq1$.