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36 Multibody Systems Approach to Vehicle Dynamics
6. The dot product {A} m • {B} m of the vectors {A} m and {B} m is defined
as a scalar whose magnitude is |A| |B| cos , being the angle between
the vectors. The dot product is evaluated in terms of a matrix product
as follows:
{A} m • {B} m {A} T m{B} m (2.56)
7. The cross product {A} m {B} m of the vectors {A} m and {B} m is defined
as a vector {C} m whose magnitude is |A||B| sin , being the angle
between {A} m and {B} m . The vector {C} m is perpendicular to the plane
containing the other two and its direction is the direction of advance of a
right-handed screw, lying parallel to {C} m , when subjected to a rotation
which would bring {A} m into alignment with {B} m by the shortest path.
The cross product is evaluated in terms of a matrix product as follows:
{C} m {A} m {B} m [A] m {B} m (2.57)
where
⎡ 0 Az
Ay ⎤
[ A] m
⎢
Az 0 Ax
⎥
⎢
⎥
⎣⎢
Ay
Ax 0 ⎦⎥
and { B}
⎡Bx⎤
⎢
By
⎥
⎢ ⎥
⎣⎢
Bz⎦⎥
[A] m is known as the skew-symmetric form of the vector {A} m .
(2.58)
8. The angle between the vectors {A} m and {B} m may be determined
from
{}
cos A T m{}
B
| A|| B|
(2.59)
9. If vectors {A} m and {B} m are parallel then {A} m can be represented
using a scalar f as follows:
{A} m f {B} m (2.60)
10. If vectors {A} m and {B} m are perpendicular then
{A} m • {B} m {A} T m{B} m 0 (2.61)
11. The scalar triple product D of the vectors {A} m , {B} m and {C} m is a
scalar defined by
D {{A} m {B} m } • {C} m (2.62)
12. The vector triple product {D} m of the vectors {A} m , {B} m and {C} m is
a vector {D} m defined by
{D} m {A} m {{B} m {C} m } (2.63)
13. If the vector {A} T [Ax Ay Az] is rotated through angle about the
x-axis of frame m then the new vector {A} m is given by
⎡1 0 0 ⎤ ⎡Ax⎤
{ A} m
⎢
0 cos sin
⎥ ⎢
Ay
⎥
⎢
⎥ ⎢ ⎥
(2.64)
⎣⎢
0 sin cos
⎦⎥
⎣⎢
Az⎦⎥
For rotation about O m Y m and O m Z m , the square matrix above is replaced
by those given in equations (2.22) and (2.23).
m