coordinates of a vector from the coordinates of two points
This type of exercise can be solved using the function : vector_coordinates
Coordinates of a vector from two points
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.
Directing vector of a line
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.
Coordinates of the midpoint of a segment
The midpoint of [AB] has coordinates `((x_(a)+x_(b))/2;(y_(a)+y_(b))/2)` in the system (O,`vec(i)`,`vec(j)`).
Distance between two points
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
.
Dot product
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.
Cross product
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.
Scalar triple product
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.
Determinant of two vectors (2x2)
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.
Determinant of three vectors (3x3)
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.