Some problems on lines in a plane

The distance between a point and a line L with normal equation can be calculated by the following formula

, .

Equation of a line passing through two given points

or

Angle between two straight lines. Let there be given a general form of equations of two lines and

, .

The problem of evaluating the angle between two lines and reduces to evaluating the angle between their normal vectors and . Thus,

.

If the straight lines and given by equations

, , or by equations

, then

. .

The lines and are parallel if or or

and are perpendicular if or or .

The plane Point–normal equation of a plane

A plane may be specified by giving its inclination and one of its points. To describe the inclination it is convenient to use a nonzero vector, which is perpendicular to a plane. This vector is called normal vector and denoted by . The equation of the plane passing through the point and having normal vector is as follows:

. (1)

This equation is called the point–normal equation of a plane.

Equation (1) may also be put in the form

, (2)

where .

The intercept equation of a plane

Let us consider a general equation of a plane where all the coefficients are not equal to zero. This equation can be rewritten in the form

, (3)

Where . This equation has rather clear geometric meaning. The numbers a, b and c are the values of intercepts with accordance signs which a plane cuts from the axes .

The normal equation of a plane

Similar to the case of straight line in a plane there is a normal equation for any plane which is completely determined by its distance p from the origin and the normal unite vector passing through the origin and directed perpendicularly to the plane. Here α, β and γ are the direction angles of this vector. Then the normal form of equation will be as follows:

. (4)

The distance between a point and a plane

Let be a given arbitrary point in space and the general equation of arbitrary plane P. Then the distance d from the point to the plane P can be calculated by the formula

. (5)

For the normal form of the equation of plane P this distance will be as follows:

. (6)

Equation of a plane passing through three given points

Just like any two points determine a unique line, any three points which don’t lie on the same line determine a unique plane. Suppose that we know such three points and determine a plane P. Then point lies in the plane P if and only if three vectors are coplanar. Therefore the triple scalar product of these vectors must be equal to zero. Using the coordinates of the vectors , , we obtain thought for equation of the plane P:

. (9)

This is the Cartesian equation of the plane .

The angle between two planes. Two planes may be parallel if and only if their normal vectors and are parallel or the following proportion are valid:

,

Two planes may be perpendicular to each other if and only if their normal vectors and are perpendicular. It implies the condition

.

Two intersecting planes determine two angles of intersection, an acute angle and its supplement . It is clear the angle between two planes is the same as the angle between their normal vectors. From the scalar product we have

.

The distance between two parallel planes and ,

is given by the formula

.