Source : ct|01.06.13

< Mathematics et physics

Vectors and vector fields

A scalar x is a real number. For example, 7, 2/3 or root of 2 are scalars. A scalar field is the set of values that takes a scalar x at different points in space.

A vector is a real number which is completed with an origin, a line, and a direction (positive or negative) with respect to this line. To draw a vector, we assigne him a length proportional to the value of the real number associated. A vector field is the set of values, lines and directions that takes a vector V at different points in space.

In mathematics, scalars and vectors are expressed without units (these are abstract elements). In contrast, in physics, the scalars and vectors have a unit (that of the concret element they represent). For example (in physics), a length of 3m, a mass of 5kg or a time of 20s are scalars. The velocity at time t (expressed in m/s) of an object located at a point x, y, z of the space is a vector.

Components of a vector

A vector V(x,y,z) placed in a fixed coordinate system (0,i,j,k) and whose components are Vx(x,y,z), Vy(x,y,z) and Vz(x,y,z) is noted

Addition of two vectors

The sum of two vectors U(x,y,z) and V(x,y,z) located in a fixed reference system (0,i,j,k) and having respectively the components Ux(x,y,z), Uy(x,y,z) Uz(x,y,z) and Vx(x,y,z), Vy(x,y,z), Vz(x,y,z) in this system gives a vector W=U+V with components Wx(x,y,z)=Ux(x,y,z)+Vx(x,y,z), Wy(x,y,z)=Uy(x,y,z)+Vy(x,y,z) and Wz(x,y,z)=Uz(x,y,z)+Vz(x,y,z).

The addition of two vectors corresponds to the notion of resultant vector. For example, in physics, a force is represented by a vector F. The addition of two forces F1 and F2 having different directions and intensities gives any resultant force F=F1+F2.

Scalar product of vectors

Consider two vectors U and V. The scalar product (U.V) of these two vectors is equal to the scalar product of the lengths of the two vectors by the cosine of the angle they form together.
U.V = U.V cosθ

The scalar product of two vectors is equal to the product of the length of the first vector by the length of the projection of the second vector on the first vector.

The scalar product corresponds to the notion of moving a vector on a directed segment. For example, in physics, the work W (J) of a force is equal to the scalar product of the vector force F (N) by the oriented distance d (m) on which this force moves.
W = F.d = F.d cosθ

With U(x, y, z)= Ux i +Uy j + Uz k and V(x, y, z)= Vx i+ Vy j + Vz k, we have U.V= (Ux i + Uy j + Uz k).(Vx i + Vy j + Vz k), therefore :
U.V= (UxVx i.i + UxVy i.j+ UxVz i.k) + (UyVx j.i + UyVy j.j+ UyVz j.k) + (UzVx k.i + UzVy k.j+ UzVz k.k)
with i.i=j.j=k.k=1 and i.j=j.k=i.k=j.i=k.i=k.j=0 because cos 0=1 and cos π/2=0

Therefore U.V = UxVx+ UxVy+ UxVz

Vector product of vectors

Consider two vectors U and V. The vector product UΛV of these two vectors is the vector W whose :

• length is equal to the product of the lengths of the two vectors by the sine of the angle they form together. W = sinθ U.V
  
• line is perpendicular to the plane formed by the vectors U and V
• the direction is such that (U,V,W) forms a direct trihedron.

The vector product of two vectors corresponds to the multiplication of the value of the first vector by the value of the second vector projected onto a axis which is perpendicular to the first vector.
The following diagram illustrates geometrically this correspondence ( in lenght, the vector product is the area of the parallelogram subtended by the U and V vectors).

With U(x, y, z)= Ux i +Uy j + Uz k et V(x, y, z)= Vx i+ Vy j + Vz k, we have UΛV= (Ux i+ Uy j + Uz k) Λ (Vx i + Vy j + Vz k), thus :
UΛV= (UxVx iΛi + UxVy iΛj+ UxVz iΛk) + (UyVx jΛi+ UyVy jΛj+ UyVz jΛk) + (UzVx kΛi+ UzVy kΛj+ UzVz kΛk)
with iΛi=jΛj =kΛk= 0, iΛj=k, jΛk=i, kΛi=j, and jΛi=-k, kΛj=-i, iΛk=-j because sin 0=0 and sin π/2=1

Thefore
UΛV= (UyVz-UzVy) i + (UzVx-UxVz) j + (UxVy-UyVx) k

To find the components of a vector product, we can write it in the form of a determinant.