Vectors: the basics

In short: A vector describes a quantity with both direction and magnitude, for example a displacement, velocity, or force. Unlike a simple number (scalar), a single number alone isn’t enough to describe a vector.

In more detail: In the plane or in space, vectors are usually written as arrows or as coordinate pairs or triples, for example (3, 4) or (1, -2, 5).

In detail

• Addition: Vectors are added component-wise, which geometrically corresponds to placing the arrows end to end.
• Scaling: Multiplying by a number stretches or shrinks the vector; a negative number reverses its direction.
• Magnitude (length): A vector’s magnitude is calculated using the Pythagorean theorem (satz-des-pythagoras-anwenden).
• Dot product: The dot product of two vectors gives a number and helps determine the angle between them.
• Unit vector: A vector with length 1 indicates only a direction, without carrying any magnitude.
• Application in physics: Forces, velocities, and accelerations are represented as vectors because they have a direction.
• Coordinate system: Vectors can be visually drawn and calculated in an x-y or x-y-z coordinate system (trigonometrie-sinus-kosinus-tangens).