Cartesian Coordinate System

17 If we consider two different sets of cartesian orthonormal coordinate systems {ei, e2, e3} and {e, e2, e3}, any vector v can be expressed in one system or the other v = vj ej = vj ej (1.30)

is To determine the relationship between the two sets of components, we consider the dot product of v with one (any) of the base vectors erv = vi = vj (erej) (1.31)

19 We can thus define the nine scalar values aj — ©î *6j — cos(xi, xj )

which arise from the dot products of base vectors as the direction cosines. (Since we have an orthonormal system, those values are nothing else than the cosines of the angles between the nine pairing of base vectors.)

20 Thus, one set of vector components can be expressed in terms of the other through a covariant transformation similar to the one of Eq. 1.27.

1.1 Vector tk-

we note that the free index in the first and second equations appear on the upper and lower index respectively.

21 Because of the orthogonality of the unit vector we have apaq = 5pq and a^an = Smn.

22 As a further illustration of the above derivation, let us consider the transformation of a vector V from (X, Y, Z) coordinate system to (x, y, z), Fig. 1.6:

Vector Calculator
Figure 1.6: Arbitrary 3D Vector Transformation

23 Eq. 1.33 would then result in

ax ax ax ax ay ay

X Jy Jz

Vx vy vz and aj is the direction cosine of axis i with respect to axis j

• ajx = (ax X, a'Y, af) direction cosines of x with respect to X, Y and Z

• a, = (ayX, a'Y,af) direction cosines of y with respect to X, Y and Z

• ajz = (azX, a]Y,af) direction cosines of z with respect to X, Y and Z

24 Finally, for the 2D case and from Fig. 1.7, the transformation matrix is written as

a2 a2

but since 7 = 2 + a, and f3 = 2 — a, then cos 7 = — sin a and cos f3 = sin a, thus the transformation matrix becomes cos a sin a sin a cos a

or z

Figure 1.7: Rotation of Orthonormal Coordinate System

Figure 1.7: Rotation of Orthonormal Coordinate System

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