How to sort vector

Any vector can be decomposed into the sum of vectors, and such possibilities are endless. The task to decompose the vector can be given as in the geometric form, and the form of formulas, this will depend on the solution of the problem.

You will need

• - initial vector;
• - vector for which you want to decompose.

Instruction

1. If you want to decompose a vector in the drawing, select the direction for parts. For convenience of calculations most often used decomposition of the vectora, parallel to the coordinate axes, but you can choose absolutely any convenient direction.
2. Draw one of the components of the vectors, in which case it must proceed from the same point as the original (the length you choose). Connect the ends of the source and the received vectorand another vectorom. Please note: two of the received vectorand the result should lead you to the same point as the source (if you move the arrows).
3. Move the resulting vectorand in the place where they are convenient to use, while maintaining the direction and length. Regardless of where the vectorand will be, in total they will be equal to the original. Please note that if you place the resulting vectorand so that they proceeded from the same point as the source, and the dotted line connecting their ends will produce a parallelogram, and the original vector will coincide with one of the diagonals.
4. If you need to decompose the vector {x1,x2,X3} with respect to the basis, that is, given the vectorof am {P1, P2, P3}, {q1,q2,q3}, {r1,r2,r3}, we proceed as follows. Substitute the coordinate values into the formula x=ar+βq+γr.
5. As a result, you will obtain a system of three equations P1A+q1β+r1γ=x1, p2α+q2β+r2γ=x2, p3α+q3β+r3γ=X3. Solve this system using method of summations or matrices, find the coefficients α, β, γ. If the task is given in the plane, the solution is more simple, because instead of three variables and equations you will receive only two (they will be in the form P1A+q1β=x1, p2α+q2β=x2). Write your answer in the form x=AP+βq+γr.
6. If you get an infinite number of solutions, make a conclusion that the vectors p, q, r lie in the same plane with the vectorω of x and decompose it in the specified way definitely not.
7. If solutions, the system has to write the answer to the problem: the vectors p, q, r lie in one plane, and the vector x in another, so it is impossible to decompose the given image.