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Kinematics and dynamics of rigid bodies 35
Differentiation of this vector with respect to time t gives
⎡AzBy ˙ AzBy ˙ AyBz ˙ AyBz
˙ ⎤
d
d ({ A } 1 { B } 1
t
) ⎢
AzBx ˙ AzBx ˙ AxBz ˙ AyBz
˙ ⎥
⎢
⎥
⎢AyBx ˙ AyBx ˙ AxBy ˙ AyBy
˙ ⎥
⎣
⎦
or
d
d ({ A } 1 { B } 1 ) { A }
t
{ B˙
} { A˙
} { B }
(2.50)
(2.51)
The result of this operation is a vector. Note again that the rule for the differentiation
of the cross product is similar to the rule for the differentiation
of the product of two scalars.
2.2.12 Summary
1. A vector is expressed in terms of the magnitudes of its three orthogonal
components listed, in natural order, as the elements of a column matrix.
The brackets { } are used as a shorthand representation of a vector. Thus
the vector {A} m with components of magnitude Ax, Ay and Az is represented
by
⎡Ax⎤
{ A} m
⎢
Ay
⎥
⎢ ⎥
⎣⎢
Az⎦⎥
(2.52)
2. The suffices m and n appended to the vector {A} m/n indicate that its components
were measured relative to the axes of frame m but that the vector
was then resolved into components parallel to the axes of frame n.
We say that the vector is referred or transformed to frame n. If m is equal
to n then the vector may be written as {A} m .
3. The magnitude of the vector {A} m is represented by |A|, or simply by A
when the meaning is unambiguous. This magnitude is given by
2 2 2
| A|
Ax Ay Az
1 1 1 1
(2.53)
4. The cosines of the angles which {A} m makes with the X, Y and Z axes
respectively of frame m are known as its direction cosines {l} m . This
vector is derived from {A} m as follows:
A
{} l m {} m
| A |
(2.54)
Since the magnitude of {l} m is unity, it is called a unit vector.
5. If {A} m , {B} m and {C} m are vectors of the same dimensions and {C} m is
the resultant of {A} m and {B} m then the equation
{C} m {A} m {B} m (2.55)
expresses the triangle law for the addition or subtraction of vectors. This
equation is only valid if all vectors are referred to the same frame.