Page 43 - Aeronautical Engineer Data Book

30 Aeronautical Engineer’s Data Book

1 1 1
2
3
–1
5
7
tan x = x – 33 x + 3 3 x – 33 x + ... (x 5 1)
3 5 7
2.8.7 Vector algebra
Vectors have direction and magnitude and
satisfy the triangle rule for addition. Quantities
such as velocity, force, and straight-line
displacements may be represented by vectors.
Three-dimensional vectors are used to repre­
, A , A
sent physical quantities in space, e.g. A x y z
or A i + A j + A k.
y
z
x
Vector Addition
,
The vector sum V of any number of vectors V 1
, V where = V a i + b j + c k, etc., is given
V 2 3 1 1 1 1
by
V = V + V + V + ... = (a + a + a + ...)i
2
1
2
3
1
3
+ b + b + ...)j + (c + c + c + ...)k
+(b 1 2 3 1 2 3
Product of a vector V by a scalar quantity s
sV = (sa)i + (sb)j + (sc)k
+ s )V = s V + s V (V + V )s = V s + V s
(s 1 2 1 2 1 2 1 2
where sV has the same direction as V, and its
magnitude is s times the magnitude of V.
Scalar product of two vectors, V ·V 2
1
V ·V = |V ||V |cos+
2
2
1
1
Vector product of two vectors, V 2 V 2
1
V 2 V |=|V ||V |sin +
2
1
1
2
where + is the angle between V and V .
2
1
Derivatives of vectors
d dB dA
33 (A · B) = A · 33 + B · 33
dt dt dt
de
If e(t) is a unit vector 33 is perpendicular to e:
dt
de
that is e · 33 = 0.
dt