How to determine if two points lies on the same line

If you are given two points, then you can safely say that they lie on the same straight, because through any two points you can draw a straight line. But how do you find out, lie all the points on the straightif points three, four, or more? To prove the belonging of points one direct in several ways.

How to determine if two points lies on the same line
How to write the equation of the line
What is the axis of symmetry
How to solve integrals

You will need

The points defined by coordinates.

Instruction

If you are given the points with coordinates (x1, Y1, z1), (x2, Y2, z2), (X3, Y3, z3), find the equation of the straight, using the coordinates of any two points, for example, first and second. To do this, substitute the appropriate values into the equation of the straight: (x-x1)/(x2-x1)=(u-U1)/(U2-U1)=(z-z1)/(z2-z1). If one of the denominators is zero, just Paranaita to zero the numerator.

Find the equation of the straight, knowing two points with coordinates (x1, Y1), (x2, Y2), is even simpler. To do this, substitute values in formula (x-x1)/(x2-x1)=(u-U1)/(U2-U1).

Having obtained the equation of the straightline that passes through two points,, substitute the values of the coordinates of the third point into it instead of the variables x and y. If equality happened is correct, then all three points lie on the same straight. Similarly, you can check the affiliation of this straight .

Check the affiliation of all points direct, checking the equality of the tangents of the angles of the connecting segments. To do this, check whether a true equality (x2-x1)/(X3-x1)=(U2-U1)/(U3-U1)=(z2-z1)/(z3-z1). If one of the denominators is zero, then a single straight should satisfy the condition x2-x1=X3-x1, Y2-Y1=Y3-Y1, z2-z1=z3-z1.

Another way to verify the ownership of the three points of direct – count the area of the triangle which they form. If all points lie on a straight, then its area will be zero. Substitute the coordinate values into the formula: S=1/2((x1-X3)(Y2-Y3)-(x2-X3)(Y1-Y3)). If after all the calculations you got zero - so the three points lie on the same straight.

To find the solution of the problem graphically, draw the coordinate plane and find the point at the specified coordinates. Then draw a line through two of them and continue to the third point, let's see if she would pass through it. Note, this method is suitable only for points in the plane with coordinates (x, y), if the point set in space and has coordinates (x, y, z), then this method is not applicable.

Is the advice useful?