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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 ° .


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.


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.


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 particle projected from origin moving on x-y plane passes through P & Q having consituents (9,7) , (18,4) respectively.find eq. of trajectry.
rahul Reply
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
Practice Key Terms 3

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