# 2.2 Vectors in three dimensions  (Page 3/14)

 Page 3 / 14

Find the distance between points ${P}_{1}=\left(1,-5,4\right)$ and ${P}_{2}=\left(4,-1,-1\right).$

$5\sqrt{2}$

Before moving on to the next section, let’s get a feel for how ${ℝ}^{3}$ differs from ${ℝ}^{2}.$ For example, in ${ℝ}^{2},$ lines that are not parallel must always intersect. This is not the case in ${ℝ}^{3}.$ For example, consider the line shown in [link] . These two lines are not parallel, nor do they intersect.

You can also have circles that are interconnected but have no points in common, as in [link] .

We have a lot more flexibility working in three dimensions than we do if we stuck with only two dimensions.

## Writing equations in ℝ 3

Now that we can represent points in space and find the distance between them, we can learn how to write equations of geometric objects such as lines, planes, and curved surfaces in ${ℝ}^{3}.$ First, we start with a simple equation. Compare the graphs of the equation $x=0$ in $ℝ,{ℝ}^{2},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{ℝ}^{3}$ ( [link] ). From these graphs, we can see the same equation can describe a point, a line, or a plane.

In space, the equation $x=0$ describes all points $\left(0,y,z\right).$ This equation defines the yz -plane. Similarly, the xy -plane contains all points of the form $\left(x,y,0\right).$ The equation $z=0$ defines the xy -plane and the equation $y=0$ describes the xz -plane ( [link] ).

Understanding the equations of the coordinate planes allows us to write an equation for any plane that is parallel to one of the coordinate planes. When a plane is parallel to the xy -plane, for example, the z -coordinate of each point in the plane has the same constant value. Only the x - and y -coordinates of points in that plane vary from point to point.

## Rule: equations of planes parallel to coordinate planes

1. The plane in space that is parallel to the xy -plane and contains point $\left(a,b,c\right)$ can be represented by the equation $z=c.$
2. The plane in space that is parallel to the xz -plane and contains point $\left(a,b,c\right)$ can be represented by the equation $y=b.$
3. The plane in space that is parallel to the yz -plane and contains point $\left(a,b,c\right)$ can be represented by the equation $x=a.$

## Writing equations of planes parallel to coordinate planes

1. Write an equation of the plane passing through point $\left(3,11,7\right)$ that is parallel to the yz -plane.
2. Find an equation of the plane passing through points $\left(6,-2,9\right),$ $\left(0,-2,4\right),$ and $\left(1,-2,-3\right).$
1. When a plane is parallel to the yz -plane, only the y - and z -coordinates may vary. The x -coordinate has the same constant value for all points in this plane, so this plane can be represented by the equation $x=3.$
2. Each of the points $\left(6,-2,9\right),$ $\left(0,-2,4\right),$ and $\left(1,-2,-3\right)$ has the same y -coordinate. This plane can be represented by the equation $y=-2.$

Write an equation of the plane passing through point $\left(1,-6,-4\right)$ that is parallel to the xy -plane.

$z=-4$

As we have seen, in ${ℝ}^{2}$ the equation $x=5$ describes the vertical line passing through point $\left(5,0\right).$ This line is parallel to the y -axis. In a natural extension, the equation $x=5$ in ${ℝ}^{3}$ describes the plane passing through point $\left(5,0,0\right),$ which is parallel to the yz -plane. Another natural extension of a familiar equation is found in the equation of a sphere.

what are the factors effecting demand sedule
we should talk about more important topics, you can search it on Google n u will find your answer we should try to focus on how we can improve our society using economics
shubham
so good night
hon
ways of improving human capital
what is human capital
kelly
Capital can be defined as man made assets use in production .
Abdulai
What is the differences between central Bank And Commercial Bank ?. 2 for each
Abdulai
Two types of bank clearing house.
Abdulai
what are the most durable assets of a bank
Ngongang
What is Opportunity Cost?
may be defined as expression of cost in terms of forgone alternative.
Abdulai
Helloo, im new, can i get to know more?
You ask questions on any topics you find difficult.
Favour
What is opportunity cost?
Cephas
is price elasticity of demand the same as elasticity of demand
not really
Victoria
hi
Gh
hello
Bhartendu
i hope everyone be ok
Gh
No
Hassan
Favour
No
William
cleophas
price elasticity of demand is the reaction of customers /demand to price changes(increase or decrease) elasticity of demand is the reaction of prices brought about by the change in demand
Victoria
thank you
Favour
state the laws of demand and supply
William
dd: when price rises demand decreases whereas when price reduces dd rises ss: when ss rises the price rises and when ss decreases price also reduces. There is a positive relationship
Dhoonah
nice
Victoria
Draw a demand curve graph
William
though price elasticity and elasticity are used interchangeably, the demand can respond to income changes and prices of related goods as well.
Gurpalak
explain the difference between merit goods and public goods and show why it is possible for profit to be made in the supply of one of these types of good but not the other
Kavishek
Public goods are defined as products where, for any given output, consumption by additional consumers does not reduce the quantity consumed by existing consumers. Merit goods are, for example, education and to some extent the health-care. They are provided by state as "good for you".
ahmed
The ladies are doing much better than the men
Blacks
what happens when there is a shift in demand curve?
Favour
What is Specialization ? Explain in detail
any one ?
specialisation is a method of production whereby an entity focuses on the production of a limited scope of goods to gain a greater degree of efficiency.
Favour
It's ok
hello
Onome
yah
Abdulai
No. price elasticity of demand refers to the manna in which price of good demanded fluctuate mean while elasticity of demand explains the way consumer change in their willingness as they plan or purchase a good
Ngongang
diffirence between demand and supply
Bonny
what is economic
It is a social science which studies human behavior as a relationship between ends and scarce which have alternative uses
Obeng
what is norminal wage
is the wages measured in money as distinct from actual purchasing power
Favour
what is demand curve
this is a curve that slop downward from left to rich
Obeng
yes
Basanta
different between capital and wealth
Wealth refers to the amount of asset you have, while, capital is the amount of cash money you have with you now and willing to invest in any business.
Favour
What is scale of reference?
What is monopoly?
Finda
It is the control of market by single seller or producer
Mayen
the exclusive possession or control of the supply or trade in a commodity or services
Brains
what is scarcity
scarcity means that the resources which we can produce goods and services relatives to wants for them.
Bonny
what is demand
demand means that's good demand according to your needs is called demand
Bonny
needs of people ar called demand
Francis
what's the difference between opportunity cost and production possibility curve?
Francis
apportunity cost means a goods which can be replace by other goods without any ease of saticfaction
Bonny
different between capital and wealth
Samuel
apportunity cost means the profit lose when one alternative is selected over other
Bonny
what is economocs
Economics is a science which studies human behaviour as a relationship between ends and scarce means which have alternative uses.
Abubakari
It deals with making choices in the face of scarcity
Abu
what is perfect complements?
Bilal
what is scarcity
Bonny
difference between fixed policy and monetary policies
Difference between extinct and extici spicies
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?