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L02_Coordinate_Transformation

L02_Coordinate_Transformation - CE507 Lecture 2 Coordinate...

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CE507 Lecture 2 Coordinate Transformation Start with a coordinate system 1 2 3 ( , , ) x x x a right-hand system with origin O. Rotate it to a new coordinate system, 1 2 3 ( , , ) x x x with the same origin O. Let , , 1 2 3 ˆ ˆ ˆ e e e be the unit vectors along the 1 2 3 , , x x x axes, and , , 1 2 3 ˆ ˆ ˆ e e e the corresponding unit vectors along the 1 2 3 , , x x x axes. Define: angle between the - (new) axis and - (old)axis. ij i j x x α = and ij a = cos ( ij α ) = ˆ ˆ i j e e ′⋅ , called the direction cosines . We will study the matrix [ ] ( ) ij a a = in this lecture. Note that a given vector, x ± , say, can have two different representations w.r.t. the two coordinate systems.
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