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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.

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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

state the properties of a base
Muhammad Reply
state 3 properties of a base
Muhammad
State three properties of a base
mubarak
a stone is projected from the top of the tower 24.5m high with velocity of 39.2m/s at an angle of 30° with horizontal compute time taken to hit he ground
Sarah Reply
it took a car driver 30mins to travel a distance of 50km.what is the average speed
Isiguzo Reply
Suppose the master cylinder in a hydraulic system is at a greater height than the cylinder it is controlling. Explain how this will affect the force produced at the cylinder that is being controlled.
Louise Reply
Why is popo less than atmospheric? Why is popo greater than pipi?
Louise
The old rubber boot shown below has two leaks. To what maximum height can the water squirt from Leak 1? How does the velocity of water emerging from Leak 2 differ from that of Leak 1? Explain your responses in terms of energy.
Louise
David rolled down the window on his car while driving on the freeway. An empty plastic bag on the floor promptly flew out the window. Explain why.
Louise Reply
the pressure differential exerted a force on the bag greater than the gravitational force holding it on the floor.
gtitboi
what is angular velocity
Sthandazile Reply
The rate of change in angular displacement is defined as angular velocity.
Manorama
a length of copper wire was measured to be 50m with an uncertainty of 1cm, the thickness of the wire was measured to be 1mm with an uncertainty of 0.01mm, using a micrometer screw gauge, calculate the of copper wire used
Nicole Reply
What is the answer please
Mustapha
it took a car driver 30 mins to travel a distance of 50km what is the average speed .
Isiguzo
If centripetal force is directed towards the center,why do you feel that you're thrown away from the center as a car goes around a curve? Explain
Maira Reply
if there is a centripetal force it means that there's also a centripetal acceleration, getting back to your question, just imagine what happens if you pull out of a car when it's quickly moving or when you try to stop when you are running fast, anyway, we notice that there's always a certain force..
Lindomar
... that tends to fight for its previous direction when you try to attribute to it an opposite one ou try to stop it.The same thing also happens whe a car goes around a curve, the car it self is designed to a"straight line"(look at the position of its tyres, mainly the back side ones), so...
Lindomar
... whenever it goes around a curve, it tends to throw away its the occupiers, it's given to the fact that it must interrupt its initial direction and take a new one.
Lindomar
Which kind of wave does wind form
Matthias Reply
calculate the distance you will travel if you mantain an average speed of 10N m/s for 40 second
Abdulai Reply
400m/s
Feng
hw to calculate the momentum of the 2000.0 elephant change hunter at a speed of 7.50 m/s
Kingsley Reply
how many cm makes 1 inches
Hassan Reply
2.5
omwoyo
2.54cm=1inche
omwoyo
how do we convert from m/s to km/hr
Toni Reply
When paddling a canoe upstream, it is wisest to travel as near to the shore as possible. When canoeing downstream, it may be best to stay near the middle. Explain why?
SANA Reply
Explain why polarization does not occur in sound
Nuradeen
one ship sailing east with a speed of 7.5m/s passes a certain point at 8am and a second ship sailing north at the same speed passed the same point at 9.30am at what distance are they closet together and what is the distance between them then
Kuber Reply
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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