Vectors

A vector;
an example: a displacement vector;
initial and terminal points of a displacement vector;
what does u=v mean?
a vector has many "representatives";
zero vector 0;
addition of vectors ("the triangle rule" and "the parallelogram rule");
"scalar"="number",
multiplication of a vector by a scalar;
example: 0v=0;
the negative -v of a vector: -v=(-1)v;
difference of vectors (defined through the other two operations: u−v=u+(−1)v);
components of a vector, examples;
position vector of a point;
magnitude (length) of a vector u=⟨u₁,u₂,u₃⟩: |u|=[(u₁)²+(u₂)²+(u₃)²]^1/2 (derivation using the Pythagorean Theorem);
the vector addition and the multiplication of a vector by a scalar in components;
fundamental properties of vectors;
properties of these "machines");
standard basis vectors i=⟨1,0,0⟩, j=⟨0,1,0⟩, k=⟨0,0,1⟩;
unit vectors (vectors of length 1);
construction of a unit vector a/|a| in the same direction as a vector a≠0, example;
application in physics: adding forces;
"machine for adding two vectors" and "machine for multiplying a vector by a scalar", constructing complicated expressions by using only these two machines.