Introduction

Up: Contents

Introduction

Vectors:

Properties:

Vectors as list of numbers:
. If the list is written horizontally, the vector is called a row vector; if it is written vertically, it is a column vector. (To distinguish the rows and columns in matrices.)
Components: r₁=x=4, r₂=y=2.
Base vectors:
.
Addition:
(4,2)+(1,3)=(4+1,2+3)=(5,5) or .
Multiplication by a scalar:
or .
Dot (scalar) product:

If the dot product is zero, the vectors are by definition orthogonal to each other. Length (norm) of a vector: . Unit vectors have length one. The distance between two points and is by definition :
The projection of onto is
Small determinants:
Cross (vector) product in 3D:

and normal to both vectors.
Line through point P parallel to vector :

This applies to any number of dimensions.
A hyperplane in Rⁿ (n-dimensional space) is the points satisfying a single scalar linear equation. In general:
i.e.

3D: A plane:
a x + b y + c z = d
2D: A line:
a x + b y = d
Curves in Rⁿ: with t the parameter. The unit tangent to the curve is .
Tangent planes to a surface F(x,y,z)=0 at a point P on the surface:
where N can be taken as the gradient of F:
Complex numbers: where . Complex numbers are special two-dimensional vectors. You get the complex conjugate of a complex number by everywhere replacing by .

Up: Contents