Chapter 4
Motion in Two and Three Dimensions
In this chapter we will continue to study the motion of objects without the
restriction we put in chapter 2 to move along a straight line. Instead we will
consider motion in a plane (two dimensional motion) and motion in space
(three dimensional motion) The
following vectors will be defined for two- and three- dimensional motion:
Displacement
Average and instantaneous velocity
Average and instantaneous acceleration
We will consider in detail projectile motion and uniform circular motion as
examples of motion in two dimensions
Finally we will consider relative motion, i.e. the transformation of velocities
between two reference systems which move with respect to each other with
constant velocity
(4 -1)

Position Vector
The position vector of a particle is defined as a vector whose tail is at
a reference point (usually the origin O) and its tip is at the particle at
point P.
The position vectoExampl r in te : he f
r
r
igure is:
ˆ
ˆ ˆ
r xi yj zk= + +
r
( )

2 2 2 2
ˆ
ˆ ˆ
r x i y j z k= + +
r
( ) ( ) ( )
2 1 2 1 2 1
ˆ ˆ
ˆ ˆ ˆ ˆ
r x x i y y j z z k xi yj zk∆ = − + − + − = ∆ + ∆ + ∆
r
(4 -3)
2 1
x x x∆ = −
2 1
y y y∆ = −
2 1
z z z∆ = −
The displacement r can then be written as:∆
r

t
t + Δt
Average and Instantaneous Velocity
Following the same approach as in chapter 2 we define the average
velocity as:
displacement
average velocity =
time interval
ˆ ˆ

1. Vector moves towards vector and 0
2. The direction of the ratio (and thus )approaches
t
avg
r r r
r
v
∆
∆ →
∆
∆
r r r
r
r
the direction
of the tangent to the path at position 1
3.
avg
v v→
r r
( )
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ
x y z
d dx dy dz
v xi yj zk i j k v i v j v k
dt dt dt dt
= + + = + + = + +
r
(4 - 5)

avg
v v v
a
t t
− ∆
= =
∆ ∆
r r r
r
We define as the instantaneous acceleration as the limit:
( )
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ
lim
0
y
x
z
x y z x y z
dv
dv
v dv d dv
a v i v j v k i j k a i a j a k
t dt dt dt dt dt
t
∆
= = = + + = + + = + +
∆
∆ →
r r