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This task of Least Square Method (LSM) is dedicated just to illustrate IMIMS in solving creative problem in Analytical Geometry with visualization. _{}^{}

Least Square Method (LSM) task is the subject of Analytic Geometry and is realized in 3D Euclidean vector space with Cartesian coordinate system and is formulated to find an orthogonal projection in 2D subspace of any vector. In general, LSM itself is a significant part of computation mathematics from the solution of differential equations (boundary value problems) to statistics and vary in realizations implied by the type of vector spaces.

To solve this task in IMIMS environment students should know the following notions and be able to apply them by performing computations with Octave software: Euclidean space, Cartesian coordinate system, vectors, subspaces and their relations to (hiper)planes, basis of space and subspace, scalar product of vectors, orthogonality, solution of linear system of equations.

The task consist of projection vector ** pr** computation in 2D subspace of any vector

To solve the task student must step by step recursively enter 5 input data. We present here the following test example.

**1-st Step**. Student chooses plane vector ** pln**=[1, -1, 1, 0] and enters it in the first input field:

[1, -1, 1, 0] then clicks the button Check Result.

Server returns a picture of coordinate system and plane.

**2-nd Step**. Student using Octave editor in IMIMS window below (or MatLab package) computes the basis, consisting of two vectors ** b1** and

[5, 5, 0]

[5, 0, -5] then clicks the button Check Result.

If basis vectors are computed correctly, server returns a picture of coordinate system, plane and two basic vectors depicted.

**3-rd Step**. Student chooses 3D vector ** v** to compute its projection on the plane:

[3, 4, 5] then clicks the button Check Result.

Server returns a picture of coordinate system, plane, two basic vectors and vector ** v** depicted.

**4-rd Step**. Student using Octave editor in IMIMS window below computes projection vector ** pr** expressed as a linear combination of two basic vectors

[16/15, -11/15] then Check Result button is clicked.

If projection vector ** pr** is computed correctly, server returns a picture of coordinate system, plane, two basic vectors, vector

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