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26 Multibody Systems Approach to Vehicle Dynamics
{B} 1
θ
{A} 1
Fig. 2.5
Vector dot product
2.2.2 The dot (scalar) product
The dot, or scalar, product {A} 1 • {B} 1 of the vectors {A} 1 and {B} 1 yields a
scalar C with magnitude equal to the product of the magnitude of each vector
and the cosine of the angle between them.
Thus:
{A} 1 • {B} 1 |C| |A| |B| cos (2.8)
The calculation of {A} 1 • {B} 1 requires the solution of
{A} 1 • {B} 1 {A} T 1 {B} 1 AxBx AyBy AzBz (2.9)
⎡Bx⎤
T
where {} B 1
⎢
By
⎥
and {} A 1 [ Ax Ay Az]
⎢ ⎥
⎣⎢
Bz⎦⎥
(2.10)
The T superscript in {A} 1 T indicates that the vector is transposed.
Clearly {A} 1 • {B} 1 {B} 1 • {A} 1 and the dot product is a commutative
operation. The physical significance of the dot product will become apparent
later but at this stage it can be seen that the angle between two vectors
{A} 1 and {B} 1 can be obtained from
cos { A}
1
{} B 1
•
| A|| B|
(2.11)
A particular case which is useful in the formulation of constraints representing
joints and the like is the situation when {A} 1 and {B} 1 are perpendicular
making cos 0.
As can be seen in Figure 2.6 the equation that enforces the perpendicularity
of the two spindles in the universal joint can be obtained from
{A} 1 • {B} 1 0 (2.12)
2.2.3 The cross (vector) product
The cross, or vector, product of two vectors, {A} 1 and {B} 1 , is another vector
{C} 1 given by
{C} 1 {A} 1 {B} 1 (2.13)
The vector {C} 1 is perpendicular to the plane containing {A} 1 and {B} 1 as
shown in Figure 2.7.