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This programme looks at the problem of specifying a vector in a three dimensional space and demonstrates a simple method of calculating the linear combination of basis vectors which will specify an...d vector. (This programme was re-edited in 1995 as FOUM469Y)
Module code and title: M203, Introduction to pure mathematics M203; 05 1980 1980 Rights owned or controlled by The Open University This material can be used in accordance with The Open University conditions of use. A link to the conditions can be found at the bottom of all OUDA web pages. 00:24:24 + Show more... Martin Wright Norman Gowar; Robin Wilson BBC Open University Basis vectors; Linear combination; Linear dependence; Linear independence Geometry, Algebraic; Linear algebraic groups; Orthogonal polynomials Norman Gowar discusses the problem of specify a vector. He explains how a vector in a two dimensional plane can be described as a linear combination of two vectors. In such a plane two linearly independent vector form a sufficient basis to describe any vector. He now examines a vector in a three dimensional plane. A full description of this requires three linearly independent vectors. Three such vectors form a basis. He then shows how two vectors can be shown to be linearly independent. This technique also applies to three vectors in a three dimensional plane. Robin Wilson now works through an example to test this result. He then solves a further equation to express a vector q as a linear combination of vectors x, y and z. This requires the resolution of three simultaneous equations. As this procedure can be tedious he looks for a simpler method of solving these problems. This relies upon the vectors x, y and z forming an orthogonal basis. Vectors are orthogonal if they are at right angles. Such vectors are also linearly independent. Robin explains how to test for when two vectors are orthogonal and then works through some examples to demonstrate the techniques Norman Gowar discusses the significance of this result. Robin Wilson then works through a further example, using the fact that the vectors are orthogonal, to specify the vector q in terms of vectors x, y and z. He then goes on to discuss the importance of the orthonormal base. This is a basis in which the vectors are orthogonal and each vector in it has length 1. Norman Gowar reviews the findings of the programme HOU2853 FOUM006N no