<< Chapter < Page Chapter >> Page >
By the end of this section, you will be able to:
  • Calculate position vectors in a multidimensional displacement problem.
  • Solve for the displacement in two or three dimensions.
  • Calculate the velocity vector given the position vector as a function of time.
  • Calculate the average velocity in multiple dimensions.

Displacement and velocity in two or three dimensions are straightforward extensions of the one-dimensional definitions. However, now they are vector quantities, so calculations with them have to follow the rules of vector algebra, not scalar algebra.

Displacement vector

To describe motion in two and three dimensions, we must first establish a coordinate system and a convention for the axes. We generally use the coordinates x , y , and z to locate a particle at point P ( x , y , z ) in three dimensions. If the particle is moving, the variables x , y , and z are functions of time ( t ):

x = x ( t ) y = y ( t ) z = z ( t ) .

The position vector    from the origin of the coordinate system to point P is r ( t ) . In unit vector notation, introduced in Coordinate Systems and Components of a Vector , r ( t ) is

r ( t ) = x ( t ) i ^ + y ( t ) j ^ + z ( t ) k ^ .

[link] shows the coordinate system and the vector to point P , where a particle could be located at a particular time t . Note the orientation of the x , y , and z axes. This orientation is called a right-handed coordinate system ( Coordinate Systems and Components of a Vector ) and it is used throughout the chapter.

An x y z coordinate system is shown, with positive x out of the page, positive y to the right, and positive z up. A point P, with coordinates x of t, y of t, and z of t is shown. All of P’s coordinates are positive. The vector r of t from the origin to P is also shown as a purple arrow. The coordinates x of t, y of t and z of t are shown as dashed lines. X of t is a segment in the x y plane, parallel to the x axis, y of t is a segment in the x y plane, parallel to the y axis, and z of t is a segment parallel to the z axis.
A three-dimensional coordinate system with a particle at position P ( x ( t ), y ( t ), z ( t )).

With our definition of the position of a particle in three-dimensional space, we can formulate the three-dimensional displacement. [link] shows a particle at time t 1 located at P 1 with position vector r ( t 1 ) . At a later time t 2 , the particle is located at P 2 with position vector r ( t 2 ) . The displacement vector     Δ r is found by subtracting r ( t 1 ) from r ( t 2 ) :

Δ r = r ( t 2 ) r ( t 1 ) .

Vector addition is discussed in Vectors . Note that this is the same operation we did in one dimension, but now the vectors are in three-dimensional space.

An x y z coordinate system is shown, with positive x out of the page, positive y to the right, and positive z up. Two points, P 1 and P 2 are shown. The vector r of t 1 from the origin to P 1 and the vector r of t 2 from the origin to P 2 are shown as purple arrows. The vector delta r is shown as a purple arrow whose tail is at P 1 and head at P 2.
The displacement Δ r = r ( t 2 ) r ( t 1 ) is the vector from P 1 to P 2 .

The following examples illustrate the concept of displacement in multiple dimensions.

Polar orbiting satellite

A satellite is in a circular polar orbit around Earth at an altitude of 400 km—meaning, it passes directly overhead at the North and South Poles. What is the magnitude and direction of the displacement vector from when it is directly over the North Pole to when it is at −45 ° latitude?


We make a picture of the problem to visualize the solution graphically. This will aid in our understanding of the displacement. We then use unit vectors to solve for the displacement.


[link] shows the surface of Earth and a circle that represents the orbit of the satellite. Although satellites are moving in three-dimensional space, they follow trajectories of ellipses, which can be graphed in two dimensions. The position vectors are drawn from the center of Earth, which we take to be the origin of the coordinate system, with the y -axis as north and the x -axis as east. The vector between them is the displacement of the satellite. We take the radius of Earth as 6370 km, so the length of each position vector is 6770 km.

An x y coordinate system, centered on the earth, is shown. Positive x is to the east and positive y to the north. A blue circle larger than and concentric with the earth is shown. Vector r of t 1 is an orange arrow from the origin to the location where the blue circle crosses the y axis (90 degrees counter clockwise from the positive x axis.) Vector r of t 2 is an orange arrow from the origin to the location on the blue circle at minus 45 degrees. Delta r vector is shown as a purple arrow pointing down and to the right, starting at the head of vector r of t 1 and ending at the head of vector r of t 2.
Two position vectors are drawn from the center of Earth, which is the origin of the coordinate system, with the y -axis as north and the x -axis as east. The vector between them is the displacement of the satellite.
Got questions? Get instant answers now!

Questions & Answers

definition of inertia
philip Reply
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
An inherent property by virtue of which the body remains in its pure state or initial state
why current is not a vector quantity , whereas it have magnitude as well as direction.
Aniket Reply
the flow of current is not current
bcoz it doesn't satisfy the algabric laws of vectors
The Electric current can be defined as the dot product of the current density and the differential cross-sectional area vector : ... So the electric current is a scalar quantity . Scalars are related to tensors by the fact that a scalar is a tensor of order or rank zero .
what is binomial theorem
Tollum Reply
hello are you ready to ask aquestion?
Saadaq Reply
what is binary operations
What is the formula to calculat parallel forces that acts in opposite direction?
Martan Reply
position, velocity and acceleration of vector
Manuel Reply
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
hello Lydia.
What is momentum
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
Ekuri Reply
I need the solving for this question
is the eye the same like the camera
I can't understand
same here please
I think the question is that ,,, the working principal of eye and camera same or not?
yes i think is same as the camera
what are the dimensions of surface tension
why is the "_" sign used for a wave to the right instead of to the left?
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
(10/6) ÷0.4=4.167 per sec
what is the formula for pressure?
Goodness Reply
force is newtom
and area is meter squared
so in SI units pressure is N/m^2
In customary United States units pressure is lb/in^2. pound per square inch
who is Newton?
John Reply
a scientist
that discovered law of motion
but who is Isaac newton?
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
Besides his work on universal gravitation (gravity), Newton developed the 3 laws of motion which form the basic principles of modern physics. His discovery of calculus led the way to more powerful methods of solving mathematical problems. His work in optics included the study of white light and
and the color spectrum
Practice Key Terms 3

Get the best University physics vol... course in your pocket!

Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?