%path = "maths/vectors" %kind = kinda["texts"] %level = 11

Vectors

What is a Vector?

A multidimensional vector can be seen as independently choosing (value) from more variables (categories, quantities, dimensions).

The values (number+unit) must be addable independently.

The units are the unit vectors. Together they form the basis and are therefore also called basis vectors.

The choice from one variable is a vector, too, a one-dimensional vector.

The whole vector can be multiplied by a number, the scalar, and yields a vector again.

Example:

  • If I go into a shop, then the products there are my vector space (coordinate system, CS) and my shopping basket is a vector, i.e. a fixing of the value (how much?) of each variable (here product).

  • If my wife went shopping, too, then the baskets add up independently at home, i.e. milk + milk, butter + butter, …

Coordinate Transformation

A matrix transforms a vector from one coordinate system to a vector of another coordinate system. Therefore we learn first about vectors. The matrix comes into play, when we want to change from one coordinate system to another.

Example

% include('r.a0',withnr=False)

How do we notate vectors?

Notation is not the vector itself.

Vector Operations

\coordinate (0) at (0,0);
\coordinate (A) at (1,3);
\coordinate (B) at (4,2);
\coordinate (C) at (2,1);
\tikzset{->}
\draw[black,very thick] (0) -- (A) node [midway,left]{$\vec{x}$};
\draw[black,very thick] (0) -- (B) node [near end,right,below]{$\vec{y}$};
\draw[black,very thin]  (0) -- (C) node [midway,right,below]{$x_y$};
\draw[-,thin] (A) -- (C) node [midway,right]{$x_{\perp y}$};

Apart from addition there are two other important vector operations.