The vector work – one of the most widespread actions used in vector algebra. This operation found wide circulation in science and technology. This concept is most evident and successful it is used in theoretical mechanics.
Instruction
1. Consider a mechanical task which solution requires the vector work. It is known that moment of force concerning the center is equal to the work of this force on her shoulder (see fig. 1a). H shoulder in the situation presented in the drawing is determined by a formula of h=|OP|sin(π-φ)= |OP| sinφ. Here F is attached to a point of River. On the other hand Fh is equal to the area of the parallelogram constructed on vectors the SHOUTING and F.
2. Force F causes rotation P relatively 0. As a result the vector directed by the known rule of "gimlet" turns out. Therefore the work Fh is the module of a vector of moment of force of OMo which is perpendicular to the plane containing vectors of F and OMo.
3. By definition vector work an and b is the vector with designated with = [and, b] (there are also other designations, most often through multiplication by "cross").s has to satisfy to the following properties: 1) with ortogonalen also b (is perpendicular) and; 2) | c |= |a| |b| sinf, where f a corner between and and b; 3) the three of vetor and, b and with right, that is the shortest turn from a to b is made counterclockwise.
4. Without going into details, it should be noted that for the vector work all arithmetic actions except property of commutativity (shift) are fair, that is [and, b] is not equal [b, a]. Geometrical meaning of the vector work: its module is equal to the area of a parallelogram (see fig. 1b).
5. Finding of the vector work soglasnopo definitions time very difficult. To solve an objective, it is convenient to use data in a coordinate form. Let in the Cartesian coordinates: a (ax, ay, az) =ax*i+ay*j+az*k, a b (bx, by, bz) =bx*i+by*j+bz*k where i, j, k are vectors-orty of coordinate axes.
6. In this case multiplication by rules of removal of brackets of algebraic expression. At the same time consider that sin(0)=0, sin(π/2)=1, sin (3π/2)=-1, the module of everyone Horta is equal 1 and the three of i, j, k right, and vectors mutually of an ortogonalna. Then receive: with = [and, b] = (ay*bz-az*by) of i-(ax*bz-az*bx) j + (ax*by-ay*bx) of k= with ((ay*bz-az*by), (az*bx-ax*bz), (ax*by-* bx)). (1) This formula is also the rule of calculation of the vector work in a coordinate form. Its shortcoming – bulkiness and, as a result, difficult memorability.
7. For simplification of a technique of calculation of the vector work use a vector - the determinant presented in figure 2. Follows from the data provided on the drawing that on the following step of disclosure of this determinant which was conducted on its first line just and there is an algorithm (1). As you can see, there are no particular problems with storing.