# 2.5 Equations of lines and planes in space  (Page 9/19)

 Page 9 / 19

$x=1+t,y=3+t,z=5+4t,$ $t\in ℝ$

a. $P\left(1,3,5\right),$ $v=⟨1,1,4⟩;$ b. $\sqrt{3}$

$\text{−}x=y+1,z=2$

Find the distance between point $A\left(-3,1,1\right)$ and the line of symmetric equations

$x=\text{−}y=\text{−}z.$

$\frac{2\sqrt{2}}{\sqrt{3}}$

Find the distance between point $A\left(4,2,5\right)$ and the line of parametric equations

$x=-1-t,y=\text{−}t,z=2,$ $t\in ℝ.$

For the following exercises, lines ${L}_{1}$ and ${L}_{2}$ are given.

1. Verify whether lines ${L}_{1}$ and ${L}_{2}$ are parallel.
2. If the lines ${L}_{1}$ and ${L}_{2}$ are parallel, then find the distance between them.

${L}_{1}:x=1+t,y=t,z=2+t,$ $t\in ℝ,$ ${L}_{2}:x-3=y-1=z-3$

a. Parallel; b. $\frac{\sqrt{2}}{\sqrt{3}}$

${L}_{1}:x=2,y=1,z=t,$ ${L}_{2}:x=1,y=1,z=2-3t,$ $t\in ℝ$

Show that the line passing through points $P\left(3,1,0\right)$ and $Q\left(1,4,-3\right)$ is perpendicular to the line with equation $x=3t,y=3+8t,z=-7+6t,$ $t\in ℝ.$

Are the lines of equations $x=-2+2t,y=-6,z=2+6t$ and $x=-1+t,y=1+t,z=t,$ $t\in ℝ,$ perpendicular to each other?

Find the point of intersection of the lines of equations $x=-2y=3z$ and $x=-5-t,y=-1+t,z=t-11,$ $t\in ℝ.$

$\left(-12,6,-4\right)$

Find the intersection point of the x -axis with the line of parametric equations

$x=10+t,y=2-2t,z=-3+3t,$ $t\in ℝ.$

For the following exercises, lines ${L}_{1}$ and ${L}_{2}$ are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting.

${L}_{1}:x=y-1=\text{−}z$ and ${L}_{2}:x-2=\text{−}y=\frac{z}{2}$

The lines are skew.

${L}_{1}:x=2t,y=0,z=3,$ $t\in ℝ$ and ${L}_{2}:x=0,y=8+s,z=7+s,$ $s\in ℝ$

${L}_{1}:x=-1+2t,y=1+3t,z=7t,$ $t\in ℝ$ and ${L}_{2}:x-1=\frac{2}{3}\left(y-4\right)=\frac{2}{7}z-2$

The lines are equal.

${L}_{1}:3x=y+1=2z$ and ${L}_{2}:x=6+2t,y=17+6t,z=9+3t,$ $t\in ℝ$

Consider line $L$ of symmetric equations $x-2=\text{−}y=\frac{z}{2}$ and point $A\left(1,1,1\right).$

1. Find parametric equations for a line parallel to $L$ that passes through point $A.$
2. Find symmetric equations of a line skew to $L$ and that passes through point $A.$
3. Find symmetric equations of a line that intersects $L$ and passes through point $A.$

a. $x=1+t,y=1-t,z=1+2t,$ $t\in ℝ;$ b. For instance, the line passing through $A$ with direction vector $\mathbf{\text{j}}:x=1,z=1;$ c. For instance, the line passing through $A$ and point $\left(2,0,0\right)$ that belongs to $L$ is a line that intersects; $L:\frac{x-1}{-1}=y-1=z-1$

Consider line $L$ of parametric equations $x=t,y=2t,z=3,$ $t\in ℝ.$

1. Find parametric equations for a line parallel to $L$ that passes through the origin.
2. Find parametric equations of a line skew to $L$ that passes through the origin.
3. Find symmetric equations of a line that intersects $L$ and passes through the origin.

For the following exercises, point $P$ and vector $\text{n}$ are given.

1. Find the scalar equation of the plane that passes through $P$ and has normal vector $\text{n}.$
2. Find the general form of the equation of the plane that passes through $P$ and has normal vector $\text{n}.$

$P\left(0,0,0\right),$ $\text{n}=3\text{i}-2\text{j}+4\text{k}$

a. $3x-2y+4z=0;$ b. $3x-2y+4z=0$

$P\left(3,2,2\right),$ $\text{n}=2\text{i}+3\text{j}-\text{k}$

$P\left(1,2,3\right),$ $\text{n}=⟨1,2,3⟩$

a. $\left(x-1\right)+2\left(y-2\right)+3\left(z-3\right)=0;$ b. $x+2y+3z-14=0$

$P\left(0,0,0\right),$ $\text{n}=⟨-3,2,-1⟩$

For the following exercises, the equation of a plane is given.

1. Find normal vector $\text{n}$ to the plane. Express $\text{n}$ using standard unit vectors.
2. Find the intersections of the plane with the axes of coordinates.
3. Sketch the plane.

[T] $4x+5y+10z-20=0$

a. $\mathbf{\text{n}}=4\text{i}+5\mathbf{\text{j}}+10\mathbf{\text{k}};$ b. $\left(5,0,0\right),$ $\left(0,4,0\right),$ and $\left(0,0,2\right);$
c.

$3x+4y-12=0$

$3x-2y+4z=0$

a. $\text{n}=3\text{i}-2\text{j}+4\text{k};$ b. $\left(0,0,0\right);$
c.

$x+z=0$

Given point $P\left(1,2,3\right)$ and vector $\text{n}=\text{i}+\text{j},$ find point $Q$ on the x -axis such that $\stackrel{\to }{PQ}$ and $\text{n}$ are orthogonal.

$\left(3,0,0\right)$

Show there is no plane perpendicular to $\text{n}=\text{i}+\text{j}$ that passes through points $P\left(1,2,3\right)$ and $Q\left(2,3,4\right).$

Find parametric equations of the line passing through point $P\left(-2,1,3\right)$ that is perpendicular to the plane of equation $2x-3y+z=7.$

$x=-2+2t,y=1-3t,z=3+t,$ $t\in ℝ$

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
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