The parallelogram is considered certain if one of its bases and side and also a corner between them are set. The problem can be solved also by methods of vector algebra (then even the drawing will not be required). At the same time the basis and side need to be set vectors and to use geometrical interpretation of the vector work. If only lengths of the parties are set – the task has no unambiguous decision.
It is required to you
- - paper;
- - handle;
- - ruler.
Instruction
1. parallelogram/b if only its emstorony/em of"" class= "" colorbox imagefield imagefield-imagelink" of" rel= "" gallery-step-images"" are known> the 1st way (geometrical). Given: the parallelogram of ABCD is set by length of the basis of |AD=a|, length of side of |AB=b| and a corner between them f (fig. 1). It is known that the area of a parallelogram is defined by expression of S= |a| h, and from ABF triangle: h=BF=ABsinf = |b| sinf. So, S= |a| |b| sinf. Example 1. Let |AD=a| =8, |AB=b| =4, f = p/6. Then S=8*4*sin(1/2)=16 of sq. piece.
2. the 2nd way (vector). The vector work is defined as a vector orthogonal to members of the work and purely geometrically (in number) coinciding with the area of the parallelogram constructed on its components. Given: the parallelogram is set by vectors of two of its parties of an and b according to fig. 1. For coincidence of data with an example 1 – let in coordinates of a (8, 0) and b (2sqrt (3, 2)). For calculation of the vector work in a coordinate form, the vector determinant is used (see fig. 2).
3. Considering that a(8,0,0), b (2sqrt (3.2),0.0) since axis 0z "looks" directly at us from the drawing plane, and vectors lie in plane 0xy. Not to be mistaken once again, rewrite result in a look: n= {nx, ny, nz }=i(aybz-azby) +j (azbx-axbz) +k(axby-aybx); and in coordinates: {nx, ny, nz } = {(aybz-azby), (azbx-axbz), (axby-aybx) }. Moreover, not to be confused with numerical examples, write out everything separately. nx=aybz-azby, ny=azbx-axbz, nz=axby-aybx. Having substituted the values which are available in a condition, receive: nx=0, ny=0, nz=16. In this case S= |nz| of =16 units of quarter.