If the line r:is in parametric form given a numeric value for the parameter t, we will obtain values for x and y. They are the coordinates of a point on r.

To verify if a point P(x₀,y₀) lies on r or not, we will substitute its coordinates into the x and the y

of the line .The point always lies on the line when the same value for t is obtained in both equations.

In this figure we have the line: r

1.- In your workbook calculate the coordinates of the points X of r, giving t the following values: 
t = 0.5  
t = -1  
t = -2.3  
Verify that your calculations are correct by changing the value of t in the buttons at the bottom of the figure. 

2.- Does the point Q(-2, 4.5) lie on r? 
To find out: 
-substitute -2 for x in the equation of r and find the value of t .
-substitute 4.5 for y in the equation of r and find the new t. If the two values of t coincide, the point Q lies on the line.

Q.x = -2 and Q.y = 4.5 in the buttons at the bottom. In the same figure you will see the values of t that you have calculated and if the point Q lies on the line or not. 

4.- By the same process as before, see if the point Q(-6,8) lies on r. 

5.- What value needs to be given to m, such that the point Q(4,m) lies on r?.

In this figure we have the line r  and any point P. The known point on r is Q(a,c) and its direction vector d(b,d). Also drawn are a line parallel and another one perpendicular to r which pass through P. 

r , which pass through  P (9,3) .

2.- Verify the result in the figure, changing the point P, dragging it with the mouse, and the line r changing the values of a, b, c and d, in the buttons at the bottom.