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.
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
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Kamara
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Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature