vector

Vectors 383
5.22 VECTOR FUNCTION
If vector r is a function of a scalar variable t, then we write
r = r()
t
If a particle is moving along a curved path then the position vector
r of the particle is a
function of t. If the component of f (t) along x-axis, y-axis, z-axis are f 1
(t), f 2
(t), f 3
(t) respectively.
Then,
—
f()
t
5.23 DIFFERENTIATION OF VECTORS
= f1() t i f2() t j f3()
t k
Let O be the origin and P be the position of a moving particle at time t.
—
Let
OP =
r
Let Q be the position of the particle at the time t + t and
the position vector of Q is OQ — = r r
—
PQ = —
OQ OP
—
= ( r r)
r r
r
is a vector. As t 0, Q tends to P and the chord
t
becomes the tangent at P.
dr r
We define
=
lim
t
0
dt t , then
dr
dt
dr
dt
Similarly,
is a vector in the direction of the tangent at P.
is also called the differential coefficient of r
d
2
dt
2
r
is the second order derivative of r .
r + r
O
with respect to ‘t’.
dr
d r
gives the velocity of the particle at P, which is along the tangent to its path. Also
dt
2
dt
gives the acceleration of the particle at P.
5.24 FORMULAE OF DIFFERENTIATION
(i) d
( F G)
d F dG (ii) d
( F )
d F d
F (U.P. I semester, Dec. 2005)
dt dt dt dt dt dt
d
dG d F d
dG d F
(iii) ( FG . ) F. . G(iv) ( F G)
F G
dt dt dt dt dt dt
d d a d b d c
(v) [ abc]
bc a c ab
dt dt
dt
dt
d d a d b dc
(vi) [ a ( b c)] ( b c)
a c a b
dt dt dt dt
The order of the functions
F,
G
is not to be changed.
Q
r
r
Tangent
P (r)
2