Geometric Vectors in Cartesian Form
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Hello there. To talk about geometric vectors in cartesian form, we have to remember some properties about linear algebra.
Given two vectors u and v, we say they are written in cartesian coordinates if they have the following notation:
[tex]\begin{gathered} u=\langle u_1,\,u_2\rangle \\ \\ v=\langle v_1,\,v_2\rangle \end{gathered}[/tex]Of course, this notation is for vectors in two dimensions, so we say that
[tex]u,v\in\mathbb{V}^2[/tex]That is the vector space with two dimensions.
We can extend this to all the plane, considering the coordinates can take all values in the real numbers, hence
[tex]u,v\in\mathbb{R}^2[/tex]And finally extend this to n-dimensions, but in this case we cannot understand it geometrically since we can, at most, geometrically represent a vector up to three dimensions
[tex]u,v\in\mathbb{R}^n[/tex]Some properties about vectors:
They are associative, that means that
[tex](u+v)+w=u+(v+w)[/tex]We have also the distributive property
[tex](u+v)\cdot w=u\cdot w+v\cdot w[/tex]Whereas
[tex]\cdot\text{ is the scalar product operator}[/tex]It also holds for cross products and other kinds of products.
They are commutative
[tex]u+v=v+u[/tex]This holds for the scalar product:
[tex]u\cdot v=v\cdot u[/tex]but it doesn't for the cross product
[tex]u\times v=-v\times u[/tex]Now, we have the geometrical view of vectors.
Say we have a point (x, y) and we want to define a vector from this point.
So we plug the tail of the vector at the origin and its tip in the point, as follows:
We can define a vector from point to point as well, but we say that they are equipollent to a vector with its tail at the origin and has the same magnitude of the vector we found.
In higher dimensions, we have
In cartesian form, we can rewrite the vectors in the following notation:
The scalar product is defined as:
[tex]u\cdot v=\langle u_1,u_2\rangle\cdot\langle v_1,v_2\rangle=u_1v_1+u_2v_2[/tex]For higher dimensions, it holds that
[tex]u\cdot v=\langle u_1,u_2,\cdots,u_n\rangle\cdot\langle v_1,v_2,\cdots,v_n\rangle=\sum_{i=1}^nu_iv_i[/tex]These are the main properties about vectors.
Related Questions
2. There were 132 students on the field trip. The students were divided into as many groups of 8 as possible. One group was smaller. How many students were in the smaller group? A 17 students B. 16 students C. 8 students D. 4 students
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So, there were 132 students divided in groups of 8.
If we divide:
So, there will be 8 groups of 16 and 1 group of 4 students. (The smaller group).
If f(x) = 2x2 + x - 3, which equation can be used to determine the zeros of the function?
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Given the function:
[tex]f(x)=2x^2+x-3[/tex]to find the zeros of the function, we have to solve the equation f(x) = 0, this means the following:
[tex]\begin{gathered} f(x)=0 \\ \Rightarrow2x^2+x-3=0 \end{gathered}[/tex]solving for 'x', we get the zeros of the function, if there are any.
what is the area of a sector bounded by a 114 arc
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Step1: Write out the given parameter
Θ=114°,r= 6ft
Step2; Write out the formula
The area of a sector is given as
[tex]\frac{\theta}{360}\times\pi r^2[/tex]Step3: substitute the parameters into the formula
[tex]\frac{114}{360}\times\pi\times6^2[/tex][tex]\begin{gathered} \frac{114}{10}\pi \\ \frac{57}{5}\pi \end{gathered}[/tex]Hence the area of the sector is (57/5)π interms o
Find the real part and the imaginary part of the following complex number. - 14 - 14/13
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Cameron is playing 9 holes of golf. He needs to score a total of at most 14 over par on the last four holes tobeat his best golf score. On the last four holes, he scores 7 over par, 1 under par, 4 over par, and 1 underpar.Part 1 out of 3Enter and find the value of an expression that gives Cameron's score for 4 holes of golf.The expression isCameron's score is✓ CheckNext
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In golf:
[tex]\begin{gathered} \text{ par corresponds to 0 on the number line} \\ \text{ under par corresponds to negative numbers on the number line} \\ \text{over par corresponds to positive numbers on the number line} \end{gathered}[/tex]Hence, in this case,
[tex]\begin{gathered} 7\text{ over par corresponds to +7 on the number line} \\ 1\text{ under par corresponds to -1 on the number line} \\ 4\text{ over par corresponds to +4 on the number line} \end{gathered}[/tex]Hence, the value of an expression that gives Cameron's score for 4 holes of golf is given by
[tex]7-1+4-1=6+4-1=10-1=9[/tex]The expression is : 7 - 1 + 4 - 1
Camerons's score is: 9
To find the measure of
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You have the following expression:
(2z - 3) + (5z - 6) = 180
in order to solvet the previous expression for z, proceed as follow:
(2z - 3) + (5z - 6) = 180 cancel out parenthesis
2z - 3 + 5z - 6 = 180 simplify like terms left side
7z - 9 = 180 add 9 both sides
7z = 180 + 9
7z = 189 divide by 7 both sides
z = 189/7
z = 27
Hence, the value of z is 27
Next, replace the values of z into the expression for the measureof angle M:
Hence, the measure of angle M is 129°
Lynn lines the bottom of her first pan with aluminum foil. The area of the rectangular piece of foil is 11 1/4 square inches. It's length is 4 and 1/2 inches. what is the width of the foil
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The area of the rectangular foil is
[tex]\begin{gathered} \text{area}=11\frac{1}{4}inches^2=\frac{45}{4}inches^2 \\ \text{length}=4\frac{1}{2}inches=\frac{9}{2}inches^2 \\ \text{width =?} \\ \text{area}=\text{length}\times width \\ \frac{45}{4}=\frac{9}{2}w \\ \text{cross multiply} \\ 90=36w \\ w=\frac{90}{36}=\frac{30}{12}=\frac{10}{4}=\frac{5}{2}\text{ inches} \\ \end{gathered}[/tex]A group of friends wants to go to the amusement park. They have no more than $225to spend on parking and admission. Parking is $5, and tickets cost $20 per person,including tax. Write and solve an inequality which can be used to determine p, thenumber of people who can go to the amusement park.3Inequality:рSubmit AnswerPrivacy Policy Terms of Service
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Answer:
Inequality: 5 + 20p ≤ 225
p ≤ 11
Explanation:
The total cost can be calculated as the sum of the parking and ticket costs. So, we can calculate the total cost as:
5 + 20p
Because 20p represents the total ticket cost for p people.
Then, this total cost should be less than or equal to 225. It means that the inequality that represents the situation is:
5 + 20p ≤ 225
Finally, we can solve the inequality by subtracting 5 from both sides as:
5 + 20p - 5 ≤ 225 - 5
20p ≤ 220
Then, divide both sides by 20, to get:
20p/20 ≤ 220/20
p ≤ 11
So, the number of people who can go to the amusement park is less than or equal to 11.
Therefore, the answers are:
Inequality: 5 + 20p ≤ 225
p ≤ 11
Round the value 23.731 g to three significant figures.Express your answer numerically using three significant figures.
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23.731 g rounded to three significant figures is 23.7 g.
Rule for significant figures:
All non-zero numbers are significant.
In 23.731, there are 5 non-zero numbers. So, there are 5 significant figures. To get number in 3 significant figures round the number at the first decimal place.
Now, the 23.731 g can be expressed in 3 significant figures as 23.7 g.
Convert: 3 days = minutes
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ANSWER
4320 minutes
EXPLANATION
To convert from days to minutes, first, we have to convert from days to hours. It is known that 1 day has 24 hours, so 3 days have,
[tex]3\text{ }days\cdot\frac{24\text{ }hours}{1\text{ }day}=72\text{ }hours[/tex]Then, we convert from hours to minutes. If 1 hour has 60 minutes,
[tex]72\text{ }hours\cdot\frac{60\text{ }minutes}{1\text{ }hour}=4320\text{ }minutes[/tex]Hence, there are 4320 minutes in 3 days.
I have no idea how to do this please help
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Answer:
Initial Value: 19,900
Value after 11 years: 7953
Explanation:
The initial value of the car is its value at t =0. Therefore, to find this initial value, we put in t = 0 to get
[tex]v(0)=19,900(0.92)^0[/tex][tex]\boxed{v\mleft(0\mright)=19,900}[/tex]Hence, the initial value is $19,900.
Now, to find the value after 11 years, we put t = 11 into the equation and get
[tex]v(11)=19,900(0.92)^{11}[/tex]which gives (rounded to the nearest dollar)
[tex]\boxed{v\mleft(11\mright)=7953}[/tex]which is our answer!
Hence, the value after 11 years is $7953.
consider the function f(x) whose second derivative is f''(x)=4x+4sin(x). if f(0)=4 and f'(0)=2, what is f(3)?
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which graph best repersents tge solution to the system of equations
y=x+4
y=3x-6
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The solution to the system of linear equation (x, y) are 5 and 9
Graph of Linear EquationsLinear equations, also known as first-order degree equations, where the highest power of the variable is one. When an equation has one variable, it is known as linear equations in one variable. If the linear equations contain two variables, then it is known as linear equations in two variables, and so on.
The solution of a linear equation in two variables is a pair of numbers, one for x and one for y which satisfies the equation. There are infinitely many solutions for a linear equation in two variables.
Therefore, every linear equation in two variables can be represented geometrically as a straight line in a coordinate plane. Points on the line are the solution of the equation. This why equations with degree one are called as linear equations. This representation of a linear equation is known as graphing of linear equations in two variables.
Using graph to solve this problem, the solution to the equations are (x, y) are 5 and 9 respectively.
Learn more on graph of linear equation here;
https://brainly.com/question/14323743
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Which of the equations below could be the equation of this parabola?5Vertex(0,0)10A. x = 2y2B. y= 2x2C. y = -2x2O D. x = -2y2
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Given:
Vertex = (0,0)
General vertex from of equation is:
[tex]y=a(x-h)^2+k[/tex][tex]\text{vertex = (h,k)}[/tex]So:
h = 0
k = 0
then equation is:
[tex]\begin{gathered} y=a(x-h)^2+k \\ h=0;k=0 \\ y=a(x-0)^2+0 \\ y=ax^2 \end{gathered}[/tex]Here value of "a" is negative because the graph is move downward . and all option give mode value is 2 then the equation of functuion:
[tex]y=-2x^2[/tex]What is the intersection of the sets C = {5, 7, 10, 13, 19) and D = {3, 9, 14, 15}?O null setO (5, 7, 9, 10, 13, 14, 15, 19}O {5, 9, 14)O {3, 19)
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We are given the following two sets C and D
C = {5, 7, 10, 13, 19}
6. Sheila simplified an expression using the following steps. Which property justifies Step 3?
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The distributive property of multiplication is represented by the following expression:
[tex]a\cdot(b+c)=a\cdot b+a\cdot c[/tex]Notice that Sheila uses distributive property to simplify the expression:
[tex]\begin{gathered} 5x+4(3+2x) \\ =5x+4\cdot3+4\cdot2x \\ =5x+12+8x \\ =13x+12 \end{gathered}[/tex]Example(-9, -2) (1,3)Find the slopeWrite in point slopeWrite in slope intercept formComplete the same three steps for 50extra points using the points(-6,7)(-3,6)
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The slope, m, of a line passing therough the points (x₁, y₁ ) and (x₂, y₂) can be calculated using the formula
[tex]m\text{ = }\frac{y_2-y_1}{x_2-x_1}[/tex]For the points (-9, -2) and (1, 3):
x₁ = -9, y₁ = -2, x₂ = 1, y₂ = 3
Substituting these points into the slope formula given above
[tex]\begin{gathered} m\text{ = }\frac{3-(-2)}{1-(-9)} \\ m\text{ = }\frac{5}{10} \\ m\text{ = }\frac{1}{2} \end{gathered}[/tex]The slope, m = 1/2
The point-slope form of the equation of a line passing through the points (x₁, y₁ ) and (x₂, y₂) can be calculated using the formula
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ y\text{ - (-2) = }\frac{1}{2}(x\text{ - (-9))} \\ y\text{ + 2 = }\frac{1}{2}(x\text{ + 9)} \end{gathered}[/tex]The slope-intercept form of the equation will be of the form y = mx + c
Reduce the point-slope form written above to the intercept-slope form
[tex]\begin{gathered} y\text{ + 2 = }\frac{1}{2}(x\text{ + 9)} \\ y\text{ + 2 = }\frac{x}{2}+\text{ }\frac{9}{2} \\ y\text{ = }\frac{x}{2}+\frac{9}{2}-2 \\ y\text{ = }\frac{1}{2}x\text{ +}\frac{5}{2} \end{gathered}[/tex]What is the solution to the following equation?
3(x-4)-5 = x - 3A. x = 12B. x=3C. x=8D. x = 7
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Given:
3(x-4)-5 = x - 3
Required:
To calculate which option is correct
Explanation:
[tex]\begin{gathered} 3(x-4)-5=x-3 \\ \\ 3x-12-5=x-3 \\ \\ 3x-17=x-3 \\ \\ 3x-x=-3+17 \\ \\ 2x=14 \\ \\ x=\frac{14}{2} \\ \\ x=7 \end{gathered}[/tex]Required answer:
Option D (x=7)
function c is defined by the equation c(n)=50+4n. it gives the monthly cost in dollars of visitiing a gym as a function of the number of visits v. find the value of c (7). show your reasoning and explain what the value means in this situation
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The cost function is defined as
[tex]c(n)=50+4n[/tex]The value of c(7) can be determined by substituting 7 for n into the function.
Therefore, c(7) becomes;
[tex]\begin{gathered} c(7)=50+4(7) \\ c(7)=50+28 \\ c(7)=78 \end{gathered}[/tex]This means the cost of 7 visits to the gym is $78.
The cost function shows that there is an amount that doesnt change and that is 50. Then there is one that varies or changes with every visit, or n. That means as the value of n increases or decreases, the total amount also increases or decreases. When n equals zero, then the total cost becomes 50. This means when there is no visit to the gym, the cost still remains $50.
Therefore, n is a variable that can determine changes in the total cost.
Answer: c(7)=78
Step-by-step explanation:
C(n)=50+4n
c(7)=50+4(7)
=50+28
=78
C(7)=78
Fill in the blank. The set {x|XS - 4.3) written in interval notation is
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The given expression is :
[tex]\mleft\lbrace x\mright|x\leq-4.3\}[/tex]In the given expression x is less than equal to - 4.3
so, it's domain will lie from - infinity to - 4.3
Thus :
[tex]\text{ Interval Notation: (-}\infty,-4.2\rbrack[/tex]Answer :
[tex]\text{ Interval Notation: (-}\infty,-4.2\rbrack[/tex]find the slope (1, 2), (-3, 3)
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Given:
The points are (1, 2), (-3, 3).
The slope is calculated as,
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ (x_1,y_1)=\mleft(1,2\mright) \\ (x_2,y_2)=(-3,3) \\ m=\frac{3-2}{-3-1} \\ m=\frac{1}{-4} \\ m=-\frac{1}{4} \end{gathered}[/tex]Answer: slope = -1/4
Hello, I need help with this precalculus homework question, please? I just need help with section D for the graph. HW Q3
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The answer would be option B
An easy way to see this is to look for the Y-intercept (when X=0)
So:
(13x + 13) / (8x +16) = 13/16 = 0.81
So, which graph has a Y intercept of approximately 0.81? The B
solve the following: a. 1/14 - 3 1/8 b. -7/2x^2y^2. + 5/2xy^2 + 3/x^2yc. -2 1/3 + 1 1/9
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First we analyze the denominators
2x^2y^2 is the greatest common factor for all three fractions, now we just divide each denominator with that factor and then multiply the answer by numerators
Then, the new fraction will be:
e)
how to compare fractions 15/8 and 12/7
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Answer:
To compare fractions 15/8 and 12/7
In order to comapare any two fractions, the denomiator of the two fractions must be same.
For the given fractions denominators are different.
To make the denomiator of the given fractions same, find the LCM of the denominators.
LCM of 8,7: 56
The fractions become,
[tex]\frac{15}{8}=\frac{15\times7}{8\times7}=\frac{105}{56}[/tex][tex]\frac{12}{7}=\frac{12\times8}{7\times8}=\frac{96}{56}[/tex]we get that,
[tex]\frac{105}{56}>\frac{96}{56}[/tex]we get,
[tex]\frac{15}{8}>\frac{12}{7}[/tex]15/8 is greater than 12/7.
Answer is: 15/8 is greater than 12/7
I want to know the answer and steps please I would appreciate it.
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ANSWER
[tex]578.05yd^2[/tex]EXPLANATION
Given;
[tex]\begin{gathered} diameter(d)=8yd \\ radius(r)=\frac{d}{2}=\frac{8}{2}=4 \\ height(h)=19yd \end{gathered}[/tex]Recall, the formula for finding the surface area of a cylinder is;
[tex]A=2\pi rh+2\pi r^2[/tex]Substituting the values;
[tex]\begin{gathered} A=2\pi rh+2\pi r^2 \\ =2\times3.14\times4\times19+2\times3.14\times4^2 \\ =477.28+100.48 \\ =578.05 \end{gathered}[/tex]Please help. I’m not sure how to do this. the options are a)1.3b)0.3c) 2.2d)0.4
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Step 1
Given;
Step 2
[tex]\begin{gathered} constant=\text{ height}\times width \\ let\text{ us use height=0.2} \\ width=2 \end{gathered}[/tex][tex]constant=0.2\times2=0.4[/tex]Answer;
[tex]0.4[/tex]The following data are the distances from the workplace (in miles) for the 5 employees of a small business.
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1. Given that the population data is : 15,5,8,2,5
• number of sample in data , ,n = 5
,• Mean = sum of sample in the data / number of sample
= (15+5+8+2+5)/5
= 35/5
Therefore mean = 7
2. Calculate varience as in the box below:
[tex]\begin{gathered} _{}\text{Varience = }\frac{1}{n}\mleft\lbrace(x_i-\vec{x}\mright)^2 \\ \text{ = }\frac{1}{5}\mleft\lbrace(7-15)^2+(7-5)^2+(7-8)^2+(7-2)^2+(7-5)^2\mright\rbrace \\ \text{ = }\frac{1}{5}\mleft\lbrace(-8^2\mright)+(-2)^2+(-1^2)+(5^2)+(2^2)\} \\ \text{ =}\frac{1}{5}\mleft\lbrace64\text{ + 4+ 1 +25+4}\mright\rbrace \\ \text{ = }\frac{1}{5}(98) \\ \text{ = }\frac{98}{5} \\ \therefore S\tan dard\text{ deviation = }\sqrt[]{varience\text{ }} \\ \text{ = }\sqrt[]{\frac{98}{5}}\text{ } \\ \text{ =4.427} \end{gathered}[/tex]• This means that Standard deviation = 4.43
[tex] {x}^{2} - [/tex]which could be the missing term in the expression if a factor of the expression is x-2ya) 2xyb) -2yc) [tex] {4y}^{2} [/tex]d)4y
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This is a difference of two squares.
If one factor is
[tex]x+2y[/tex]An the other is
[tex]x-2y[/tex]We have that the expression is:
[tex](x+2y)\cdot(x-2y)=x^2-4y^2[/tex]So the missing term is 4y², option c
X 0 1 | 2 3 4 y 7 15 23 31 39
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step 1
Find the slope
we take the points
(0,7) and (1,15)
m=(15-7)/(1-0)
m=8/1
m=8
step 2
Find the equation in slope intercept form
y=mx+b
we have
m=8
b=7 -------> (0,7) is the y-intercept
substitute
y=8x+7A line is drawn over this rectangle . Is the line a line of symmetry?
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Answer:
The line is not a line of symmetry because the two parts are not an exact match when you fold the rectangle over the line.
Explanation:
A line of symmetry is a line that divides the figure into two equal parts, so when you fold the figure over the line, the two parts will match exactly. So, taking into account the figure, the line drawn is not a line of symmetry.
The answer is
The line is not a line of symmetry because the two parts are not an exact match when you fold the rectangle over the line.