Geometria

	4. Application of vectors to metric problems: DISTANCES
	Block :Geometry

4.3. Distance between two points

The distance between two points P(x₁,y₁), Q(x₂,y₂) is the modulus of the vector PQ
dist(P,Q) = \|PQ\| =

For example to find the distance between the points P(3,-1) and Q(-1,2) we make the following calculations

dist[(3,-1),(-1,2)]=

1.- In your workbook calculate the coordinates of the vector PQ being P(3,-5) and Q(1,4)

2.-Now calculate the distance PQ.

Now, using the mouse, you can move the points P and Q, or change their coordinates using the buttons at the bottom. You can see the length of PQ, for different points.

3.- Verify the result in this figure.

4.- What are the coordinates of the vector PQ if P(1,4) and Q(3,-5)Now what is the distance between P(1,4) and Q(3,-5)?.

4.4. Distance from a point to a line

The distance from the point P(a,b) to the line r:Ax+By+C = 0 is:

dist(P,r) =

1.- In the previous figure the distance from the point P(-5,8) to the line
r: 2x -6y + 7 = 0 is calculated. Calculate it yourself applying the formula and test your result.

By using the mouse to move point P, or by changing its coordinates using the buttons at the bottom, and changing the coefficients in the equation of the line r, A, B and C, you can see the distance from P to r, in each case. You can see that it will always be the segment drawn from P, perpendicular to r.

2.- Calculate the distance from P(2,-1) to r: x - 3y + 5 = 0, and verify it in the previous figure.

3.- Calculate the distance from P(7,0) to r:

(firstly you must put r into its implicit form).
Verify your result in the previous figure.

4.- If you calculate the distance from P(3,3) to r: y = 2x - 3 (firstly you must put r into its implicit form), you will see that the result is equal to zero. Why does this occur? Look at it in the figure.

	Ángela Núñez Castaín

	Ministry of Education , Social Afairs and Sport. Year 2001

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