Answer:
a) The nonzero vector orthogonal to the plane through the points P, Q and R is (0, 10, -10).
b) The area of the triangle PQR is [tex]5\sqrt{2}[/tex].
Step-by-step explanation:
The statement is incomplete. The complete question will be presented below:
Consider the points below: P(2, 0, 2), Q(−2, 1, 3), R(4, 2, 4)
(a) Find a nonzero vector orthogonal to the plane through the points
P, Q, and R. (b) Find the area of the triangle PQR.
a) Let [tex]P(x,y,z) = (2,0,2)[/tex], [tex]Q(x,y,z) = (-2,1,3)[/tex] and [tex]R(x,y,z) = (4,2,4)[/tex]. First, we construct vectors PQ and PR by using the following formulas:
[tex]\overrightarrow{PQ} = Q(x,y,z) -P(x,y,z)[/tex] (1)
[tex]\overrightarrow{PQ} = (-2,1,3)-(2,0,2)[/tex]
[tex]\overrightarrow{PQ} = (-4,1,1)[/tex]
[tex]\overrightarrow{PR} = R(x,y,z) -P(x,y,z)[/tex] (2)
[tex]\overrightarrow{PR} = (4,2,4)-(2,0,2)[/tex]
[tex]\overrightarrow{PR} = (2,2,2)[/tex]
Then, we can find the nonzero vector orthogonal to the plane by means of the following cross product:
[tex]\vec S = \overrightarrow{PQ}\,\times\,\overrightarrow{PR}[/tex]
Which can be solved by the following determinant (Law of Sarrus):
[tex]\vec S = \left|\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\-4&1&1\\2&2&2\end{array} \right |[/tex]
[tex]\vec S = (2-2,2+8, -8-2)[/tex]
[tex]\vec S = (0, 10,-10)[/tex]
The nonzero vector orthogonal to the plane through the points P, Q and R is (0, 10, -10).
b) The area of the triangle is determined by the following formula:
[tex]A = \frac{1}{2}\cdot \|\overrightarrow{PQ}\,\times\,\overrightarrow{PR} \|[/tex]
[tex]A = \frac{1}{2}\cdot \|\vec S\|[/tex] (3)
[tex]A = \frac{1}{2}\cdot \sqrt{0^{2}+10^{2}+(-10)^{2}}[/tex]
[tex]A = 5\sqrt{2}[/tex]
The area of the triangle PQR is [tex]5\sqrt{2}[/tex].
Help me ASAP! Giving brainliest so pleaseeeeeee help me!
Answer:
the answer is 2, 5, 9 :))) hope i helped
A plant can grow A inches in B months where a is a non-negative rational number and B is a natural number. the height is proportional to the number of months after the plant is planted. in which to situations would be the constant proportionality be less than one for the equation equals a=kb
Soz I do not know the answer but if you need help I would recommend going to a friend or family member for help if you are still stuck you should consider messaging a teacher via email but if this is urgent and a teacher does not respond you should try to contact him/her using a phone call you should also try getting in contact with other people in the learning facility for example a T.A
Mike worked to earn $1293.12. if he worked for 32 hours, how much money does Mike earn per hour? $
Blank 1:
Answer:
$40.40
Step-by-step explanation:
a boat is tied to a dock with a 25-foot rope.when the rope is pulled tight, the boat us 18 feet away from the base of the dock. the rope connects to the boat at the level of the water. what is the measure, to the nearest tenth of a degree, of x, the angle formed by the rope and the surface of the water.
Answer:
≈ 44.0°Step-by-step explanation:
Use trigonometry
The ratio of the adjacent side of a right triangle to the hypotenuse is called the cosine:
cos x = 18/25 = 0.72cos x = 0.72x = arccos 0.72x = 43.9455° ≈ 44.0° roundedSolve the question in the photo!!!!!!!!!!!!!
A negative divided by a positive = negative
A positive divided by a negative = negative
A negative divided by a negative = positive
The answer is D
Answer:
The answer is d
Step-by-step explanation:
Hope this helps
What fraction of 400 gallons is 180 L? Give your answer in simplest form. 1 gallon = 4.5 L.
Answer:
400 gallons are 1100 / 63 of 126 L.
Step-by-step explanation:
Given
1 Gallon = 4.5 L
To determine
What fraction of 400 gallons is 180 L
as
1 Gallon = 4.5 L
so
400 Gallons = 400 × 4.5
= 1800 L
In order to determine the fraction of 400 gallons is 180 L, just divide 1800 by 180, so
Fraction = 2200 ÷ 126
= 1100 ÷ 63
Therefore, 400 gallons are 1100 / 63 of 126 L.
i need help with this one pls
Answer:
x=35, y=35
Step-by-step explanation:
To be honest, you don't even have to do any maths to figure this one out if you know the theory.
Basically, any angles standing on the same arc (like x and y) are equal. So we know x=y, and the first option(x=35,y=35) is the only option where x is the same as y.
John spends $2.75 on donuts for his office employees every weekday. On Sunday mornings he takes his wife and kids to the donut shop and spends $12.99. How much money does John spend on donuts throughout the week?
How much money does John spend on donuts throughout the week?
Answer:
(d)- $26.74
Step-by-step explanation:
John spends $2.75 on donuts for his office employees every weekday. On Sunday mornings he takes his wife and kids to the donut shop and spends $12.99. How much money does John spend on donuts throughout the week?
a) $23.24
b) $19.64
c) $29.14
d) $26.74
The total money spends on donuts throughout the week is $26.74 option (d) is correct.
What is an arithmetic operation?It is defined as the operation in which we do the addition of numbers, subtraction, multiplication, and division. It has a basic four operators that is +, -, ×, and ÷.
The missing options are:
a) $23.24b) $19.64c) $29.14d) $26.74We have:
John spends $2.75 on donuts for his office employees every weekday. On Sunday mornings he takes his wife and kids to the donut shop and spends $12.99.
Let the total number of working days in a week be 5
Total money spend on employees = 5×2.75 = $13.75
Total money spend on donuts throughout the week = 13.75 + 12.99
= $26.74
Thus, the total money spends on donuts throughout the week is $26.74 option (d) is correct.
Learn more about the arithmetic operation here:
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if x=2y-5 and z=16y^3, what is z in terms of x
Answer:
z in terms of x is [tex]\mathbf{z=16\left(\frac{x+5}{2}\right)^3}[/tex]
Step-by-step explanation:
We are given:
[tex]x=2y-5\: and\: z=16y^3[/tex]
We need to find z in terms of x
First we can find value of y from x =2y-5
[tex]x =2y-5\\x+5 = 2y\\y=\frac{x+5}{2}[/tex]
Now, we can put value of y in [tex]z=16y^3[/tex] to find z in terms of x
[tex]z=16y^3\\Put\:y=\frac{x+5}{2}\\z=16(\frac {x+5}{2})^3[/tex]
So, z in terms of x is [tex]\mathbf{z=16\left(\frac{x+5}{2}\right)^3}[/tex]
which set of points on the graph represents equivalent ratios.
Answer:
Points B,C and D
Step-by-step explanation:
Hope I helped :)
Answer:
Hello and (B) is correct and I would like to say GREAT JOB for the one above me.
Step-by-step explanation:
Your welcome and have a nice day. No bad day. :)
A sofa-set was brought for Rs.10000. Its value depreciated at the rate of 10% per annum. Find its value after one year
Answer:
Value after one year = Rs 9000
Step-by-step explanation:
The depreciation refers to a reduction in value of the asset (sofa-set) due to usage and other factors. The depreciation charge at the rate of 10% means that the reduction in value at 10% of cost per year.
Depreciation per year = 10000 * 0.1 = Rs 1000
So, the value after one year will be,
Value after one year = 10000 - 1000
Value after one year = Rs 9000
You've gone to fruit stand to get some fresh produce. You
notice that the person in front of you gets 5 apples and 4 oranges
for 10 dollars. You get 5 apples and 5 oranges for 11 dollars.
Create a system of equations that represents this situation.
Step-by-step explanation:
Let $x be the price of an apple and $y be the price of an orange.
5x + 4y = 10
5x + 5y = 11
btw the solution is
y = 1
x = 6/5
The length of a rectangle is
2
meters less than
3
times the width. The perimeter is
44
meters. Find the width.
Answer:
They did not state exactly what the width or length is. So in this case, let's name the number of width to be x. And thus we can derive to the conclusion of,
Step-by-step explanation:
Width : x
Length : x-44 (from the 1st sentence of question)
So, We all know that a rectangle has 2 Lengths and 2 widths. So we can derive the conclusion that,
2(x)+2(x-44)=150150
2x+2x-88=150150
4x-88= 150150
4x= 150150+88=150238
4x=150238
x= 150238÷4
=37559.5
Answer:
The width is 8 meters and the length is 3(8)-2=22 meters. Hope this helps (owo)/ brainlist please.
Step-by-step explanation:
Let x= the width
then 3x-2= length (because the length is 2 meters less than 3 times the width)
P=2l+2w
60=2(lenght)+2(width)
60=2(3x-2)+2x
60=6x-4+2x Distribute
60=8x-4 Combine like terms
64=8x Divide both sides by 8
8=x
Kendall is making and selling necklaces. It costs Kendall $2.00 to make each necklace. If she sells 40 necklaces for $5.00 each, how much profit did she make?
Answer:
Kendall makes 120$ profit by selling necklaces.
Step-by-step explanation:
To show how much it costs to make all the necklaces in total multiply 2 x 40 to get 80$. To show much much money she gets for the necklaces multiply 5 x 40 to get 200$. To show how much profit she makes subtract 80$ from 200$ to get 120$.
please help only 10 minutes
Answer:
1 gallon = 4 quarts
20 gallon = (4×20) quarts=80quarts
C. is the right answer.Brenna took a history test. She got 34 out of 85 problems right. What
percentage did Brenna get right?
Answer:40%
Step-by-step explanation:
Answer: When you convert the fraction to a percentage it is 40% Hope that this helps you
Step-by-step explanation:
y = 2x – 4 and y = x – 2
need to solve and show work i have a low grade and its only the second 9 weeks
Step-by-step explanation:
x=2
y=0
.................
write the algebraic expression for: Kim has 7 more than 5 times the amount Sylvia has.
Answer:
k = 5s + 7
Step-by-step explanation:
We will say that Kim = K and that Sylvia = S.
[tex]k=5s + 7[/tex]
Answer:
7+5s
:)
Step-by-step explanation:
please hurry i need an answer
Answer:
[tex]-\frac{11}{6}\ \frac{inches}{second}[/tex]
Step-by-step explanation:
Rate of Change
The average velocity is defined as the rate of change of the distance with respect to time. The table shows the distances of an object as a function of time.
It's required to find the average velocity between 8 s and 20 s.
The rate of change can be approximated as the slope between two points. We take from the table the points (8,32) and (20,10) and calculate the slope.
The slope can be calculated with the equation:
[tex]\displaystyle v=\frac{d_2-d_1}{t_2-t_1}[/tex]
[tex]\displaystyle v=\frac{10-32}{20-8}=\frac{-22}{12}[/tex]
Simplifying:
[tex]\displaystyle v=-\frac{11}{6}[/tex]
Choice: B. [tex]\mathbf{-\frac{11}{6}\ \frac{inches}{second}}[/tex]
The selling price of an item is $600. It is marked down by 10%, but this sale price is still marked up
from the cost of $450. Find the markup from cost to sale price.
The markup is $ (Round to the nearest dollar as needed.)
Step-by-step explanation:
Sale price
= Marked down by 10% from selling price
= 90% of selling price
= 0.9 * $600
= $540.
Sale price - Cost price
= $540 - $450 = $90.
The markup from cost to sale is $90.
The markup from cost to sale price will be $90.
What is the percentage?The amount of any product is given as though it was a proportion of a hundred. The ratio can be expressed as a quarter of 100. The phrase % translates to one hundred percent. It is symbolized by the character '%'.
The percentage is given as,
Percentage (P) = [Final value - Initial value] / Initial value x 100
The selling cost of a thing is $600. It is discounted by 10%, however, this deal cost is as yet increased from the expense of $450.
The sale price is given as,
Sale price = 0.90 x $600
Sale price = $540
The markup cost to sale will be given as,
⇒ $540 - $450
⇒ $90
The markup from cost to sale price will be $90.
More about the percentage link is given below.
https://brainly.com/question/8011401
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A submarine was at the surface of the water. 21 seconds later, it was at a depth of 294 meters. What was the average change in the submarines depth each second?
Step-by-step explanation:
21 seconds = -294 meters
=> 1 second = -294/21 = -14 meters
Hence the average change is -14 meters per second.
The average change in the submarines depth is -14 meters per second.
What is arithmetic?Arithmetic is the branch of mathematics that deals with the study of numbers using various operations on them. Basic operations of math are addition, subtraction, multiplication and division. These operations are denoted by the given symbols.
Given:
A submarine was at the surface of the water.
21 seconds later, it was at a depth of 294 meters.
We have to find the average change in the submarines depth each second.
According to given question we have
21 seconds = -294 meters
=> 1 second = -294/21 = -14 meters
Hence ,the average change in the submarines depth is -14 meters per second.
Learn more details about arithmetic here:
https://brainly.com/question/11559160
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John found the markers he wanted at two stores. At the grocery store, a pack of 5 markers cost $2.45. At the office supply store, they cost $5.00. What is the minimum number of markers needed in the pack at the office supply store for that package to be a better deal? Answer Choices : 9 10 11 12
Answer: i think so 10 because if you add 2.45+2.45+2.45+2.45= 9.8 and 10 is the nearest number to 9.8
HOPE THIS HELPS
what is 1 3/5 as a percentage?
Answer: 60%?
Step-by-step explanation:
A small measuring spoon holds 5 milliliters. How many spoons would 1 liters of liquid fill??
Answer:
200
Step-by-step explanation:
Answer:200
Step-by-step explanation:
I did the quiz and its correct
What is the equation of the line in slope-intercept form?
Answer:
The equation of the line in slope-intercept form is:
[tex]y = x + 4[/tex]
Step-by-step explanation:
The slope-intercept form of the line equation
y = mx+b
where m is the slope and b is the y-intercept
Given the equation
(1, 5) (2, 6)Determining the slope between (1, 5) and (2, 6)
(x₁, y₁) = (1, 5)
(x₂, y₂) = (2, 6)
Using the formula
Slope = m = [y₂ - y₁] / [x₂ - x₁]
= [5 - 4] / [1 - 0]
= 1 / 1
= 1
Thus, the slope of the line = m = 1
We know that the value of the y-intercept can be determined by setting x = 0 and determining the corresponding value of y.
From the graph, it is clear
at x = 0, y = 4
Thus, the y-intercept b = 4
now substituting b = 4 and m = 1 in the slope-intercept form
[tex]y = mx + b[/tex]
[tex]y = (1)x + 4[/tex]
[tex]y = x + 4[/tex]
Therefore, the equation of the line in slope-intercept form is:
[tex]y = x + 4[/tex]
!!!! PLEASE PLEASE HELP ME!!! PLEASE THANK YOU!! DOESNT NEED TO BE PERFECT!!!
Write an equation using comparison operators for more than or equal to twenty.
Answer:
>=20
Step-by-step explanation:
Given f () = (5 – 1)? what is the value of f(11)?
Record your answer and fill in the bubbles on your answer document.
Keypad
-168
1
2
3
4
6
7
00
9
0
CLEAR
Answer:
f(11) = 9
Step-by-step explanation:
f(x) = 1/4(5-x)^2
f(x) = 1/4 (5-11) ^2
f(11) = 1.25 - 2.75 ^2
f(11) = 9
The point of intersection of a system of equations is (3,4). Which of the following could represent one of the equations in the system of equations?
Select one:
A. 2x+3y=15
B. x+y=8
C. −2x+6y=18
D. −4x−y=−8
help
Answer:
c
Step-by-step explanation:
here
x=3
y=4
so,
-2×3+6×4=18
or,-6+24=18
Answer:
The answer is C
Step-by-step explanation:
Please help me answer this with the correct answer :)
Explain your answer
I will give braisnlt :)
Answer:
from the graph, there is point (1,3) and (-2,-3).
[tex]slope, \: m = \frac{(y _{2} - y _{1})}{(x _{2} - x _{1}) } \\ m = \frac{( - 3 - 3)}{( - 2 - 1)} \\ m = \frac{ - 6}{ - 3} \\ m = 2 \\ from \: general \: equation \: of \: a \: line \\ y = mx + c \\ consider \: point \: (1,3) \\ 3 = (2)(1) + c \\ 3 = 2 + c \\ c = 3 - 2 \\ c = 1 \\ substitute \: for \: m \: and \: c \: in \: y = mx + c \\ y = 2x + 1[/tex]
Answer: D => y=2x+1
I dont understand the second question in the picture
Answer:
ako naiintindihan mobako
