Let(O,`vec(i)`,`vec(j)`) be a reference frame of the plane. Let B and F be two points of coordinates `(1,15)` and `(12,2)` respectively in this frame, compute the coordinates of the middle of the segment [BF].

Let (O, `vec (i)`, `vec (j)`) a system, A and B two points which are respective coordinates (`x_a`,`y_(a)`) and (`x_(b)`,`y_(b)`) in
the system (O,`vec(i)`,`vec(j)`) .

The coordinates of the vector `vec(AB)` are (`x_(b)`-`x_(a)`,`y_(b)`-`y_(a)`) in the system (O,`vec(i)`,`vec(j)`).

The
vector coordinate calculator
allows you to do this type of calculation.

If, in a system, a line D has equation `y=m*x+p` then the vector `vecu(1;m)` is a **directing vector** of D.

The midpoint of [AB] has coordinates `((x_(a)+x_(b))/2;(y_(a)+y_(b))/2)` in the system (O,`vec(i)`,`vec(j)`).

The plane is provided with an orthonormal system (O,`vec(i)`,`vec(j)`) .
If A and B are two points with coordinates (`x_(a)`,`y_(a)`) and (`x_(b)`,`y_(b)`) in the (O,`vec(i)`,`vec(j)`) system,
then the **distance** AB is equal to:

AB=`sqrt((x_(b)-x_(a))^2+(y_(b)-y_(a))^2)`, the distance AB is also the **norm of the vector** `vec(AB)`, hich can be calculated using the
vector norm calculator
.

In the plan, in an orthonormal system `(O,vec(i),vec(j))` ,
`vec(u)` is a vector of coordinates (x,y) and `vec(v)` is a vector of coordinates (x',y'),
the **dot product** is given by the formula
xx'+yy'.

The
dot product calculator
allows this type of calculation for n-dimensional vectors.

In an orthonormal coordinate system (O,`vec(i)`,`vec(j)`,`vec(k)`), the **cross product**
of vectors `vec(u)(x,y,z)` and `vec(v)(x',y',z')` has coordinates `(yz'-zy',zx'-xz',xy'-yx')`, it notes `vec(u)^^vec(v)`.

This product can be determined using
cross product calculator.

The **scalar triple product** of three vectors `(vec(u),vec(v),vec(w))` is the number `vec(u)^^vec(v).vec(w)`.
In other words, the **scalar triple product** is obtained by calculating the
cross product
of `vec(u)` and `vec(v)` noted `vec(u)^^vec(v)`, then performing the dot product
dot product
of the vector `vec(u)^^vec(v)` and the vector `vec(w)`.
It can be calculated using the
scalar triple product calculator.

In an orthonormal coordinate system (O,`vec(i)`,`vec(j)`) , the vector `vec(u)` has coordinates (x,y)
(`vec(i)`,`vec(j)`), the vector `vec(v)` has coordinates (x',y'). The **determinant** of `vec(u)` et `vec(v)` is given by the
**formula xx'-yy'**.

This
example shows a calculation of the determinant of the vectors [[3;12];[45;2]] performed with the 2x2 determinant calculator.

Note: When the determinant of two vectors is zero, the two vectors are collinear.

In an orthonormal coordinate system (O,`vec(i)`,`vec(j)`,`vec(k)`), the vector `vec(u)` has coordinates (x,y,z)
, the vector `vec(v)` has coordinates (x',y',z'), the vector `vec(k)` has coordinates (x'',y'',z'').
The **determinant** of `vec(u)`, `vec(v)`, `vec(k)` is given by the **formula xy'z''+x'y''z+x''yz'-xy''z'-x'yz''-x''y'z**.