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Kinematics and dynamics of rigid bodies 27
{A} 1
{B} 1
Fig. 2.6
Application of the dot product to enforce perpendicularity
{C} 1
{B} 1
{A} 1
θ
Fig. 2.7
Vector cross product
The magnitude of {C} 1 is defined as
|C| |A| |B| sin (2.14)
The direction of {C} 1 is defined by a positive rotation about {C} 1 rotating
{A} 1 into line with {B} 1 . The calculation of {A} 1 {B} 1 requires {A} 1 to
be arranged in skew-symmetric form as follows:
⎡ 0
{} C 1 { A} 1 { B} 1 [ A]
1{}
B 1
⎢
Az
⎢
⎣⎢
Ay
Multiplying this out would give the vector {C} 1 :
⎡AzBy
AyBz⎤
{} C 1 ⎢
AzBx AxBz
⎥
⎢
⎥
⎣⎢
AyBx
AxBy⎦⎥
Az
Ay ⎤
Ax
⎥
⎥
0 ⎦⎥
(2.15)
(2.16)
Exchange of {A} 1 and {B} 1 will show that the cross product operation is not
commutative and that
{A} 1 {B} 1 {B} 1 {A} 1 (2.17)
0
Ax
⎡Bx⎤
⎢
By
⎥
⎢ ⎥
⎣⎢
Bz⎦⎥