Two-dimensional Vector Space with ordered Basis (i, j)Lehman - Pesonen1. Ordered Basis (i, j) in the 2-dimensional planeWe take an ordered basis E := (i, j) in the Plane, that is; two non-parallel (non-collinear) vectors i and j, whose order we keep fixed. |
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Unit AreaIn this first Sketch the ideas of Basis and Unit Area are introduced.The parallelogram generated ('spanned') by the unit vectors is always defined, but can be of area zero. We take the area of the parallelogram as the Unit Area of value 1.
Question 1
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2. Vectors in the two-dimensional Vector Space with ordered Basis (i, j)
We have a fixed ordered basis
E := (i, j)
in the Plane.
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Basis spans the vector setIn this Sketch the basis E is 'arbitrary'. All vectors ares spanned by the two base vectors.For a vector u, the coordinates with respect to the basis E are (a,c).
Question 2
How to calculate with vectors, see the axioms in Definition of General Vector Space |
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3. Signed area of parallelogram defined by two vectors in a two-dimensional Vector Space with ordered Basis E = (i, j)We have a fixed ordered basis E = (i, j) in the Plane and two vectors u and v. They define a parallelogram, whose signed area we introduce now. |
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Signed AreaIn this Sketch the basis E is 'arbitrary', but fixed. The picture is supposed to show how the Unit Area and the vector parallelogram (signed) area are tied together.
Question 3
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4. How the determinant describes area in orthonormal (rectangular unit vectors) basis caseEuclid - Lehman - PesonenWe have an ordered basis E := {i,j} in the Plane, and two vectors u and v with coordinates (a,c) and (b,d) in the basis E. |
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Determinant of two vectorsIn this Sketch the basis E is given. The picture is supposed to show how the determinant ties the unit area and the vector parallelogram together. |
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