By Robert C. Stern

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**Extra info for America’s Fighters of the 1980s: F-16 and F-18 (Warbirds Illustrated 17)**

**Example text**

Since (even if we had deﬁned positive rotations of these angles) the cosine is an even function. So the same transformation is ⎧ ⎫ ⎡ ⎤⎧ ⎫ cos θx2 x1 cos θy2 x1 cos θz2 x1 ⎨vx2 ⎬ ⎨vx1 ⎬ v = ⎣cos θx2 y1 cos θy2 y1 cos θz2 y1 ⎦ vy2 ⎩ y1 ⎭ ⎩ ⎭ v z1 cos θx2 z1 cos θy2 z1 cos θz2 z1 v z2 Obviously the columns of T1,2 are the rows of T2,1 (and have unit length as well). This leads us to another nice property of these transformation matrices, that the inverse of the direction cosine matrix is equal to its transpose, −1 T = T2,1 T1,2 = T2,1 Coordinate System Transformations 21 −1 T Since T2,1 T2,1 = T2,1 T2,1 = I3 , the 3 × 3 identity matrix, it must be true that the scalar (dot) product of any row of T2,1 with any other row must be zero.

The omission of a superscript suggests the inertial reference frame. The vector is represented in a body-ﬁxed coordinate system. {ωB }B : 14 Aircraft Flight Dynamics and Control Once deﬁned it may be convenient to omit the subscripts and superscripts of a vector, especially if the vector is referred to repeatedly, or if the notation becomes cumbersome. 1 Latitude and longitude Position on the Earth is measured by latitude and longitude. Denote latitude by the symbol λ and longitude by the symbol μ.

Waterlines are measured along the z-axis from this plane. The x − z plane is called the fuselage station plane. Fuselage stations (coordinates along the y-axis) are numbered beginning at the nose of the aircraft, not the origin. Thus, a component may be mounted on a bulkhead at Fuselage Station (FS) 300, meaning the bulkhead is 300 inches back from the nose. The y − z plane is the plane of symmetry, and coordinates along the x-axis (spanwise) are called buttock, or butt lines. Liming describes the origin of the system as simply the intersection of the plane of symmetry (which is pretty well deﬁned) and the other two planes (which are not very well Coordinate Systems 15 deﬁned).