An Approach to Martix & Linear Algebra

by Frank C. Fung ( 1st published in November, 2004. )

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Section 3 : Briefing on { orthogonal unit-vector Bases }

Topic I - Introduction & Summary for this Section :

In this section, we shall go thru the mechanics of rotating a set of { orthogonal unit-vector Basis } in a { 3-Dimensional Vector-Space } , to arrive at a desired / pre-determined orientation.

The [ Degrees-Of-Freedom ] ( DOF's ) associated with { orthogonal unit-vector Bases } is also analyzed.

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Topic II - Setting up the { Global Reference Frame } :

Let us first set up the { Global Reference Frame } , as identified by the three (3) orthogonal unit-vectors, , , and respectively, which shall remain fixed and constant throughout our analysis in this Section, as follows :

&

( Again, we present the right-hand-side diagram above for better visualization purposes ).

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Topic III - Defining a set of { orthogonal unit-vector Basis } :

In general, there are three (3) { Degrees Of Freedom } ( DOF's ) associated with identifying any vector in a { 3-Dimensional Vector-Space } ,

• but this is reduced to two (2) { Degrees Of Freedom } ( DOF's ) for a { unit-vector } , due to the { unity-constraint } placed on length of the { unit-vector } .

We can then use the angle & to identify the { unit-vector } , as per this diagram below :

We then define a set of { orthogonal unit-vector Basis } in a { 3-Dimensional Vector-Space } as :

• a set of three (3) { orthogonal unit-vectors } in that { 3-Dimensional Vector-Space } ,

  so that any vector can then be described relative to the said set of { orthogonal unit-vector Basis } .

Once we have defined one of the three (3) { unit-vectors } in a set of { orthogonal unit-vector Basis } , via the two angles & respectively,

• the positions of the other two (2) { orthogonal unit-vectors } , on the plane orthogonal to the { 1st unit-vector } , may then be defined via the 3rd { Degree Of Freedom } , using the angle .

We shall next demonstrate how to handle the angles , , & for a set of { orthogonal unit-vector Basis } .

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Topic IV - Setting up { orthogonal unit-vector Bases } :

Let us now set up a set of { orthogonal unit-vector Basis } in the { 3-Dimensional Vector-Space } , consisting of the { unit-vectors } , & , respectively :

&

• which, initially, maps exactly onto the { unit-vectors } , & respectively, as shown above.

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Topic V - The Rotation Process :

The { Rotation Process } shall consist of three (3) rotations :

The 1st Rotation :

We first rotate the set of { orthogonal unit-vector Basis } about the { axis } , to arrive at a new set of { orthogonal unit-vector Basis } , i.e. { , , } , as per the diagram below :

We then have this relation for the two (2) sets of { orthogonal unit-vector Bases } :

The 2nd Rotation :

We then rotate the set of { orthogonal unit-vector Basis } about the { axis } to arrive at a new set of { orthogonal unit-vector Basis } , i.e. { , , } , as per the diagram below :

We then have this relation for the two (2) sets of { orthogonal unit-vector Bases } :

The 3rd Rotation :

We then rotate the set of { orthogonal unit-vector Basis } about the { axis } to arrive at a new set of { orthogonal unit-vector Basis } , i.e. { , , } , as per the diagram below :

We then have this relation for the two (2) sets of { orthogonal unit-vector Bases } :

We can then combine these three (3) equations into one (1) single equation :

yielding :

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Topic VI - An Important Property :

We note that this equation above is in the general format :

The matrix in the middle, the { Matrix TAU } , shall be known as :

• a { unit-basis-on-unit-basis mapping matrix } ,

  throughout our paper here.

And we note here a very important property for this type of matices, namely :

and consequently :

This arises from :

• firstly, the { unity constraint } on the length of a { unit-vector } :

  

  

  

• and secondly, the [ orthogonality ] of the { unit-vectors } :

  

  

  

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Topic VII - An Analysis on { Degrees of Freedom } :

We then have this table for the { Degrees Of Freedom } associated with setting up a set of { orthogonal unit-vector Basis } in a { 'N'-Dimensional Vector-Space } , based on our discussion in { Topic III } above :

Then, in general, the { Degrees Of Freedom } associated with setting up a set of { orthogonal unit-vector Basis } in { 'N'-Dimensional Vector-Space } is then given by :

• { N * ( N - 1 ) / 2 }

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go to the next Section : Section 4 - Understanding the { Linear Mapping } Relation

return to FCF's Matrix & Linear Algebra HomePage

original dated 2004-11-15