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In unit vector notation, the position vectors are

r ( t 1 ) = 6770. km j ^ r ( t 2 ) = 6770. km ( cos 45 ° ) i ^ + 6770. km ( sin ( −45 ° ) ) j ^ .

Evaluating the sine and cosine, we have

r ( t 1 ) = 6770. j ^ r ( t 2 ) = 4787 i ^ 4787 j ^ .

Now we can find Δ r , the displacement of the satellite:

Δ r = r ( t 2 ) r ( t 1 ) = 4787 i ^ 11,557 j ^ .

The magnitude of the displacement is | Δ r | = ( 4787 ) 2 + ( −11,557 ) 2 = 12,509 km . The angle the displacement makes with the x- axis is θ = tan −1 ( −11,557 4787 ) = −67.5 ° .

Significance

Plotting the displacement gives information and meaning to the unit vector solution to the problem. When plotting the displacement, we need to include its components as well as its magnitude and the angle it makes with a chosen axis—in this case, the x -axis ( [link] ).

An x y coordinate system is shown. Positive x is to the east and positive y to the north. Vector delta r sub x points east and has magnitude 4797 kilometers. Vector delta r sub y points south and has magnitude 11,557 kilometers. Vector delta r points to the southeast, starting at the tail of delta r sub x and ending at the head of delta r sub y and has magnitude 12,509 kilometers.
Displacement vector with components, angle, and magnitude.

Note that the satellite took a curved path along its circular orbit to get from its initial position to its final position in this example. It also could have traveled 4787 km east, then 11,557 km south to arrive at the same location. Both of these paths are longer than the length of the displacement vector. In fact, the displacement vector gives the shortest path between two points in one, two, or three dimensions.

Many applications in physics can have a series of displacements, as discussed in the previous chapter. The total displacement is the sum of the individual displacements, only this time, we need to be careful, because we are adding vectors. We illustrate this concept with an example of Brownian motion.

Brownian motion

Brownian motion is a chaotic random motion of particles suspended in a fluid, resulting from collisions with the molecules of the fluid. This motion is three-dimensional. The displacements in numerical order of a particle undergoing Brownian motion could look like the following, in micrometers ( [link] ):

Δ r 1 = 2.0 i ^ + j ^ + 3.0 k ^ Δ r 2 = i ^ + 3.0 k ^ Δ r 3 = 4.0 i ^ 2.0 j ^ + k ^ Δ r 4 = −3.0 i ^ + j ^ + 2.0 k ^ .

What is the total displacement of the particle from the origin?

An x y z coordinate system with distances measured in micrometers and ranging from -10 to +10 micrometers is shown. The displacements delta r sub 1 equals 2 I hat plus j hat plus 2 k hat, delta r sub 2 equals -1 I hat plus 3 k hat, and delta r sub 3 equals -3 I hat plus j hat plus 2 k hat are shown as blue line segments. Vector r 1 hat starts at the origin. Each subsequent displacement starts where the previous one ends. Vector delta r total is shown as a red line starting at the origin and ending at the end of vector delta r 4. Delta r total equals 2 I hat plus 0 y hat plus 9 k hat.
Trajectory of a particle undergoing random displacements of Brownian motion. The total displacement is shown in red.

Solution

We form the sum of the displacements and add them as vectors:

Δ r Total = Δ r i = Δ r 1 + Δ r 2 + Δ r 3 + Δ r 4 = ( 2.0 1.0 + 4.0 3.0 ) i ^ + ( 1.0 + 0 2.0 + 1.0 ) j ^ + ( 3.0 + 3.0 + 1.0 + 2.0 ) k ^ = 2.0 i ^ + 0 j ^ + 9.0 k ^ μ m .

To complete the solution, we express the displacement as a magnitude and direction,

| Δ r Total | = 2.0 2 + 0 2 + 9.0 2 = 9.2 μ m, θ = tan −1 ( 9 2 ) = 77 ° ,

with respect to the x -axis in the xz- plane.

Significance

From the figure we can see the magnitude of the total displacement is less than the sum of the magnitudes of the individual displacements.

Got questions? Get instant answers now!

Velocity vector

In the previous chapter we found the instantaneous velocity by calculating the derivative of the position function with respect to time. We can do the same operation in two and three dimensions, but we use vectors. The instantaneous velocity vector    is now

v ( t ) = lim Δ t 0 r ( t + Δ t ) r ( t ) Δ t = d r d t .

Let’s look at the relative orientation of the position vector and velocity vector graphically. In [link] we show the vectors r ( t ) and r ( t + Δ t ) , which give the position of a particle moving along a path represented by the gray line. As Δ t goes to zero, the velocity vector, given by [link] , becomes tangent to the path of the particle at time t .

Questions & Answers

A central force is given as F vector (r),where a=2NM².Assuming the potential energy at infinity to be zero,calculate the potential energy of a particle located at the point (3,4)
Isa Reply
what is a vector
Honest Reply
A vector is any physical quantity which has a magnitude as well as a direction associated to it. Which means a vector is some physical quantity which has magnitude and direction.
malayala
what is matter
Seth
nice
Faith
What is the equation illustrating Williamsons ether synthesis
Queen Reply
What is the equation illustrating Williamsons ether synthesis
Kingdom
what is simple harmonic motion
Faith Reply
examples: vibrating prongs of a tuning fork and a guittar string.
Salman
It is a repetitive periodic motion of a system about an equilibrium position
Felix
SHM is the repitition process of to and fro motion.
Younus
SHM is the motion in which the restoring force is directly proportional to the displacement of body from its mean position and is opposite in direction to the displacement. From Hooke's law F=-kx
Kushal
SHM is the motion in which the restoring force is directly proportional to the displacement of body from its mean position and is opposite in direction to the displacement. From Hooke's law F=-kx
Kushal
what is a wave?
DAVID Reply
show that coefficient of friction of solid block inclined at an angle is equivalent to trignometric tangent of angle
DAVID
Wave is the transfer of energy due to the periodic vibration of the particles in the medium.
Kushal
wave is the transfer of energy
Vindora
Wave is the transfer of particles in a fluid or any way.
Younus
thanks for that definition.
Dodou Reply
Hi everyone please can dere be motion without force?
Lafon
no...
Enyia
Thanks
Lafon
hi
Omomaro
whats is schrodinger equation
Omomaro
l went spiral spring
Xalat
what is position?
Adhar Reply
position is simply where you are or where you were
Shii
position is the location of an object with respect to a two or three dimensional axes or space.
Bamidele
Can dere be motion without force?
Lafon
what is the law of homogeinity?
auson Reply
two electric lines of force never interested each other. why?
Sujit Reply
if two electric lines of force intersect eachother then their will be two tangent at a point which represent the two forces which is impossible.
Amar
proof that for BBC lattice structure 4r\root 5 and find Apf for the BBC structure
Eric Reply
what is physics?
Abdulaziz Reply
physics is deine as the specific measrument of of volume, area,nd distances...
Olakojo
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia
yes
Alex
proof that for a BBC lattice structure a= 4r/ root 5 find the APF for the BBC structure
Eric
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia Reply
tell me conceptual quetions of mechanics
Syeda Reply
I want to solve a physical question
ahmed
ok
PUBG
a displacement vector has a magnitude of 1.62km and point due north . another displacement vector B has a magnitude of 2.48 km and points due east.determine the magnitude and direction of (a) a+ b and (b) a_ b
Kou Reply
quantum
George
a+b=2.9
SUNJO
a+b
Yekeen
use Pythogorous
Dhritwan
A student opens a 12kgs door by applying a constant force of 40N at a perpendicular distance of 0.9m from the hinges. if the door is 2.0m high and 1.0m wide determine the magnitude of the angular acceleration of the door. ( assume that the door rotates freely on its hinges.) please assist me to d
Mike
what is conditions met to produce shm
Enocy Reply
what is shm
Manzoor
shm?
Grant
Why is Maxwell saying that light is an electromagnetic wave?
Bong
1st condition; It(th e BBC's system) must have some inertia which will enable it to possess Kinetic energy 2. must be able to store potential energy
Calleb
I meant "the system" not the BBC'S....."
Calleb
what a answer bro
Manzoor
kindly tell us the name of your university
Manzoor
GUlam Ishaq Khan INSTITUTE of engineering science
ali
Department of Environment Ionian University Zante Greece
Practice Key Terms 3

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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