Moduie 2. Section 1. Analitical geometry

Lecture 2.1. Straight line and plane

The straight line

General concepts

Analytical geometry combines algebra with geometry to solve geometrical problems. A point in a plane is identified with its two coordinates (x, y) and a set of points with a certain properties (a locus) is adequate to a certain relation between these coordinates x and y (the equation of the locus). As loci we will first consider geometrical curves in a plane. For example, the set of points located at a given distance R from a given point derives a circle with the equation in Cartesian coordinates

or .

Definition 1. The equation is said to be the equation of a curve L if the Cartesian coordinates of any point in the Oxy–plane satisfies this equation if and only if the point belongs to the curve L.

Then the points of intersection of two loci and (their common points) are considered as the solution of the system of equations

.

In many applications a curve traced by the motion of a particle (or a point) can be described by expressing the coordinates x and y of a point on the curve as functions of a third variable, say the time t, for a specified domain of that variable.

Definition 2. The two equations are called parametric equations, and the variable t is called a parameter.

For example, the equations

,

represents the circle of radius R whose centre is at the origin.

Different forms of the equation of a straight line

General form

, (1)

where is called a normal vector of a straight line.

Theorem 1. The curve with equation (1) is a straight line and conversely, any straight line has the first order equation.

Proof. Let be any point in Oxy-plane and L be a straight line passing through a given perpendicular to a given vector . Then the vector is perpendicular to the vector and the scalar product equals zero:

(2)

or where .

Equation (2) is called a point–normal form of the equation of a straight line.

Intercept form

If , then the equation

, (3)

where is x–intercept, and is y–intercept is called the intercept form of the equation of a straight line.

Vector form and parametric form

Let be the straight line passing through the point and parallel to the nonzero vector (direction vector). Let be any arbitrary point in Oxy-plane. Consider two vectors and .

Then the straight line L is the locus with the property or . What gives the vector equation of a line

, (4)

and parametrical equation

. (5)

Point direction (standard) form of the equation of a straight line

Finding the parameter t from both equations (5) and equating we get the point direction equation of the straight line L, passing through the point parallel to the vector :

. (6)

Slope y–intercept form

Let α be the angle of inclination of a line in the plane with respect to the x-axis, and k its slope , and b its y-intercept then equation

(7)

is called the slope y–intercept equation of the line.

Normal form

Let be the unite vector from the origin directed perpendicularly towards the straight line L and makes an angle with the positive Ox–axis. Therefore . Additionally let p be the distance from the origin to the line. Then for any point of the line is true the equation . It implies the equation

. (8)

witch is called a normal form equation of a straight line.