Vector Definition

Description of the definition and drawing of different vectors


Vectors describe a direction and magnitude in a coordinate system. For calculations we use them as lists of numbers. But for understanding it can help to display them as geometric objects.

The vector \(\underline{x}=\left[\matrix{a\\b}\right]\) has a magniutude \(|\underline{x}|=\sqrt{a^2+b^2}\)

A vector is uniquely determined by length, direction and orientation.

Plot vectors

To draw a vector, use a coordinate system. Draw a horizontal line and a line perpendicular. The horizontal line is the X-axis; the vertical line is the Y-axis. The axes labeled with numbers for the scales of the corresponding units.

The following figure shows the vector \(\overrightarrow{a}=\left[\matrix{x\\y}\right]\) or \(\overrightarrow{a}=\left[\matrix{2\\3}\right]\)

Die obere Zahl ist die x-Koordinate und die untere die y-Koordinate des Vektors.

Um den Vektor (2, 3) zu zeichen, ziehen Sie von einem Ausgangspunkt eine Linie 2 Einheiten auf der x-Achse nach rechts und 3 Einheiten auf der y-Achse nach oben

The vector is uniquely defined by its direction and its length.

Equal Vectors

The following figure shows parallel vectors of equal length, direction and orientation. Since the location of a vector is arbitrary, these vectors are equal.

Opposite Vectors

Parallel vectors of equal length but opposite in orientation are called opposite vector.

Parallel Vectors

Two vectors are called in parallel if they have the same direction. They can be different lengths and have opposite orientations.