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

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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.

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Joseph
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no can't
Lohitha
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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
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da
Application of nanotechnology in medicine
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Kamaluddeen
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narayan
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ya I also want to know the raman spectra
Bhagvanji
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what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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What is meant by 'nano scale'?
What is STMs full form?
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scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
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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?