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**vector**a, parallel to the coordinate axes, but you can choose absolutely any convenient direction.2

Draw one of the components of

**the vector**s, 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**vector**and another**vector**om. Please note: two of the received**vector**and the result should lead you to the same point as the source (if you move the arrows).3

Move the resulting

**vector**and in the place where they are convenient to use, while maintaining the direction and length. Regardless of where**the vector**and will be, in total they will be equal to the original. Please note that if you place the resulting**vector**and 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 vector**of 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 vector**s p, q, r lie in the same plane with**the vector**ω 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 vector**s p, q, r lie in one plane, and**the vector**x in another, so it is impossible to decompose the given image.