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 ):
[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.
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
located at
with position vector
At a later time
the particle is located at
with position vector
. The
displacement vector
is found by subtracting
from
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.
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
latitude?
Strategy
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.
Solution
[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.
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost
6% more than previous year? Provide with reasoning.
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Good day
How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200.
Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
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Sekou
ma management ho
Amisha
ahile becheclor ho
Amisha
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ani k kaam grnu hunxa tw
Amisha
belatari
Amisha
1st year ho
Amisha
nd u
Amisha
ahh
Amisha
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Amisha
ys
Amisha
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Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9
no Mpc is greater than 1
Y=100+.9Y+50
Y-.9Y=150
0.1Y/0.1=150/0.1
Y=1500
Kalombe
Mercy is it clear?😋
Kalombe
hi can someone help me on this question
If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output?
discuss your answer using appropriate graph.