Perpendicular

A solder is in the geometry of a straight line which is perpendicular to a given straight line or plane. Depending on whether it is a straight line or a distance from this line is also called Perpendicular or solder Recke. The intersection point of the solder with the given line or plane is called the nadir point. The solder can be constructed geometrically in various ways by ruler and compass. Are calculated, it may by determining the normal vector of the straight line or plane or by orthogonal projection of a point outside the line or plane. The length of the solder Recke is then just the distance ( perpendicular distance ) of a point from a straight line or plane.

• 3.1 Erection of the solder
• 3.2 cases of Lots
• 3.3 Example

Definition

A line is perpendicular to a straight line or plane, when

Applies, so if it is perpendicular to the straight line or plane, and thus forming a right angle therewith. The nadir point is then the intersection or the solder with the straight line or plane.

Geometric constructions

In two dimensions, allows the solder construct Just a simple way to a ruler and compass. Depending on whether a given point on the line or outside is, one speaks of the building or from the felling of the solder.

Erection of the solder

Is a given point on the line, it finds the Perpendicular passing through this point as follows. Man stabs the circle in the point and determined by drawing a circle any two points on the straight line of equally spaced. Then you increase the angle of the circle, stabs him in each one of the two points found on the straight and finds by drawing two arcs a period ( of two possible ) outside of the line equidistant from the two points. The straight line connecting the point of equal distance to the starting point, the Perpendicular is then to be carried.

Cases, the solder

If a point outside the line given, then you find the solder through this point on the line as follows. Man stabs the circle in the point and determined by drawing a circle with a correspondingly large radius of two points on the straight line with equal spacing. Then one shrinks where appropriate, the angle of the circle, stabs him in each one of the two points found on the straight and finds by drawing two arcs another point equidistant from the two points. The straight line connecting these two points of equal distance to each other, which is Perpendicular to by then and the intersection of this line with the nadir point.

An alternative design of a given point, the solder on a straight line to cut is precisely pierce the circle at two arbitrary points and in which and are each the circle that passes through the given point inscribe. These two circles then intersect at another point outside the line and the line connecting and is then the Perpendicular by. This construction can also be used for reflection.

Calculation

In analytic geometry points in the Euclidean plane or Euclidean space using the Cartesian coordinate system by position vectors

Described. Lines in the plane are typically used as a line equation in parametric form

Given, the position vector of a line point, the direction vector of the line and a real parameter. Planes in space are typically used as a plane equation in parametric form

Optionally, wherein and are real parameters, as well as and the tension vectors of the plane may not be collinear. Two vectors and in the plane or in space form a right angle, if their dot product is.

Erection of the solder

The direction vector of the perpendicular straight line to a given straight line or plane, the normal vector of the straight line or plane. Obtained in the two-dimensional case, a normal vector on a line through interchanging the two components of its direction vector and by reversing the sign of one of the two components

A normal vector of a plane, you can, if it is not given in normal form, by using the cross product of the tension vectors

Calculate. Is now a point on the straight line or plane given, then the line equation of the Perpendicular

Where is a real number. A straight line in space has no excellent normal direction, instead it has at each point a straight vertical plane whose normal vector is equal to the direction vector of the line.

Cases, the solder

If a point outside the line or plane given, then one obtains the nadir point of the solder on the straight or level orthogonal

It is also possible to precipitate the solder on a point in space onto a straight line in space. If the direction vector of the line, then you get through the nadir point

The nadir point is the one straight line or plane point, the distance is minimal. It thus defines the distance from to the line or plane as the length of the solder Recke.

Example

Consider the plane with the base point and the tension vectors and. A normal vector of the plane is then

Or even easier. The Perpendicular through the point on the plane is thus

Is now the point outside the plane given, then one obtains the nadir point of the solder onto the plane as

The distance of the point from the plane so that