# 2.5 Equations of lines and planes in space  (Page 10/19)

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Find symmetric equations of the line passing through point $P\left(2,5,4\right)$ that is perpendicular to the plane of equation $2x+3y-5z=0.$

Show that line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$ is parallel to plane $x-2y+z=6.$

Find the real number $\alpha$ such that the line of parametric equations $x=t,y=2-t,z=3+t,$ $t\in ℝ$ is parallel to the plane of equation $\alpha x+5y+z-10=0.$

For the following exercises, points $P,Q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}R$ are given.

1. Find the general equation of the plane passing through $P,Q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}R.$
2. Write the vector equation $\text{n}·\stackrel{\to }{PS}=0$ of the plane at a., where $S\left(x,y,z\right)$ is an arbitrary point of the plane.
3. Find parametric equations of the line passing through the origin that is perpendicular to the plane passing through $P,Q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}R.$

$P\left(1,1,1\right),Q\left(2,4,3\right),$ and $R\left(-1,-2,-1\right)$

a. $-2y+3z-1=0;$ b. $⟨0,-2,3⟩·⟨x-1,y-1,z-1⟩=0;$ c. $x=0,y=-2t,z=3t,$ $t\in ℝ$

$P\left(-2,1,4\right),Q\left(3,1,3\right),$ and $R\left(-2,1,0\right)$

Consider the planes of equations $x+y+z=1$ and $x+z=0.$

1. Show that the planes intersect.
2. Find symmetric equations of the line passing through point $P\left(1,4,6\right)$ that is parallel to the line of intersection of the planes.

a. Answers may vary; b. $\frac{x-1}{1}=\frac{z-6}{-1},y=4$

Consider the planes of equations $\text{−}y+z-2=0$ and $x-y=0.$

1. Show that the planes intersect.
2. Find parametric equations of the line passing through point $P\left(-8,0,2\right)$ that is parallel to the line of intersection of the planes.

Find the scalar equation of the plane that passes through point $P\left(-1,2,1\right)$ and is perpendicular to the line of intersection of planes $x+y-z-2=0$ and $2x-y+3z-1=0.$

$2x-5y-3z+15=0$

Find the general equation of the plane that passes through the origin and is perpendicular to the line of intersection of planes $\text{−}x+y+2=0$ and $z-3=0.$

Determine whether the line of parametric equations $x=1+2t,y=-2t,z=2+t,$ $t\in ℝ$ intersects the plane with equation $3x+4y+6z-7=0.$ If it does intersect, find the point of intersection.

The line intersects the plane at point $P\left(-3,4,0\right).$

Determine whether the line of parametric equations $x=5,y=4-t,z=2t,$ $t\in ℝ$ intersects the plane with equation $2x-y+z=5.$ If it does intersect, find the point of intersection.

Find the distance from point $P\left(1,5,-4\right)$ to the plane of equation $3x-y+2z-6=0.$

$\frac{16}{\sqrt{14}}$

Find the distance from point $P\left(1,-2,3\right)$ to the plane of equation $\left(x-3\right)+2\left(y+1\right)-4z=0.$

For the following exercises, the equations of two planes are given.

1. Determine whether the planes are parallel, orthogonal, or neither.
2. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer.

[T] $x+y+z=0,$ $2x-y+z-7=0$

a. The planes are neither parallel nor orthogonal; b. $62\text{°}$

$5x-3y+z=4,$ $x+4y+7z=1$

$x-5y-z=1,$ $5x-25y-5z=-3$

a. The planes are parallel.

[T] $x-3y+6z=4,$ $5x+y-z=4$

Show that the lines of equations $x=t,y=1+t,z=2+t,$ $t\in ℝ\text{,}$ and $\frac{x}{2}=\frac{y-1}{3}=z-3$ are skew, and find the distance between them.

$\frac{1}{\sqrt{6}}$

Show that the lines of equations $x=-1+t,y=-2+t,z=3t,$ $t\in ℝ,$ and $x=5+s,y=-8+2s,z=7s,$ $s\in ℝ$ are skew, and find the distance between them.

Consider point $C\left(-3,2,4\right)$ and the plane of equation $2x+4y-3z=8.$

1. Find the radius of the sphere with center $C$ tangent to the given plane.
2. Find point P of tangency.

a. $\frac{18}{\sqrt{29}};$ b. $P\left(-\frac{51}{29},\frac{130}{29},\frac{62}{29}\right)$

Consider the plane of equation $x-y-z-8=0.$

1. Find the equation of the sphere with center $C$ at the origin that is tangent to the given plane.
2. Find parametric equations of the line passing through the origin and the point of tangency.

Two children are playing with a ball. The girl throws the ball to the boy. The ball travels in

the air, curves $3$ ft to the right, and falls $5$ ft away from the girl (see the following figure). If the plane that contains the trajectory of the ball is perpendicular to the ground, find its equation.

$4x-3y=0$

[T] John allocates $d$ dollars to consume monthly three goods of prices $a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c.$ In this context, the budget equation is defined as $ax+by+cz=d,$ where $x\ge 0,y\ge 0,$ and $z\ge 0$ represent the number of items bought from each of the goods. The budget set is given by $\left\{\left(x,y,z\right)|ax+by+cz\le d,x\ge 0,y\ge 0,z\ge 0\right\},$ and the budget plane is the part of the plane of equation $ax+by+cz=d$ for which $x\ge 0,y\ge 0,$ and $z\ge 0.$ Consider $a=\text{}8,$ $b=\text{}5,$ $c=\text{}10,$ and $d=\text{}500.$

1. Use a CAS to graph the budget set and budget plane.
2. For $z=25,$ find the new budget equation and graph the budget set in the same system of coordinates.

[T] Consider $\text{r}\left(t\right)=⟨\text{sin}\phantom{\rule{0.2em}{0ex}}t,\text{cos}\phantom{\rule{0.2em}{0ex}}t,2t⟩$ the position vector of a particle at time $t\in \left[0,3\right],$ where the components of r are expressed in centimeters and time is measured in seconds. Let $\stackrel{\to }{OP}$ be the position vector of the particle after $1$ sec.

1. Determine the velocity vector $\text{v}\left(1\right)$ of the particle after $1$ sec.
2. Find the scalar equation of the plane that is perpendicular to $\text{v}\left(1\right)$ and passes through point $P.$ This plane is called the normal plane to the path of the particle at point $P.$
3. Use a CAS to visualize the path of the particle along with the velocity vector and normal plane at point $P.$

a. $\text{v}\left(1\right)=⟨\text{cos}\phantom{\rule{0.2em}{0ex}}1,\text{−}\text{sin}\phantom{\rule{0.2em}{0ex}}1,2⟩;$ b. $\left(\text{cos}\phantom{\rule{0.2em}{0ex}}1\right)\left(x-\text{sin}\phantom{\rule{0.2em}{0ex}}1\right)-\left(\text{sin}\phantom{\rule{0.2em}{0ex}}1\right)\left(y-\text{cos}\phantom{\rule{0.2em}{0ex}}1\right)+2\left(z-2\right)=0;$
c.

[T] A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) $A\left(8,0,0\right),$ $B\left(8,18,0\right),$ $C\left(0,18,8\right),$ and $D\left(0,0,8\right)$ (see the following figure).

1. Find the general form of the equation of the plane that contains the solar panel by using points $A,B,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}C,$ and show that its normal vector is equivalent to $\stackrel{\to }{AB}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{AD}.$
2. Find parametric equations of line ${L}_{1}$ that passes through the center of the solar panel and has direction vector $\text{s}=\frac{1}{\sqrt{3}}\text{i}+\frac{1}{\sqrt{3}}\text{j}+\frac{1}{\sqrt{3}}\text{k},$ which points toward the position of the Sun at a particular time of day.
3. Find symmetric equations of line ${L}_{2}$ that passes through the center of the solar panel and is perpendicular to it.
4. Determine the angle of elevation of the Sun above the solar panel by using the angle between lines ${L}_{1}$ and ${L}_{2}.$

find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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