A wagon is being pulled by a 300-pound force that makes a 30° angle with the ground . What are the horizontal and vertical components of the force vector ?

Main Answer:The linear combination of v = (13/14)u1 + (2/7)u2 + (47/14)u3  

Supporting Question and Answer:

How can we express a vector as a linear combination of  vectors using a system of equations?

To express a vector as a linear combination of  vectors using a system of equations, we need to find the coefficients that multiply each given vector to obtain the desired vector. This can be done by setting up a system of equations, where each equation corresponds to the components of the vectors involved.

Body of the Solution:

(a) To show that the vectors u1 = (2, 0, 3), u2 = (-3, 0, 2), and u3 = (0, 7, 0) form an orthogonal basis for R^3, we need to demonstrate two conditions: orthogonality and linear independence.

u1 · u2 = (2)(-3) + (0)(0) + (3)(2) = -6 + 0 + 6 = 0

u1 · u3 = (2)(0) + (0)(7) + (3)(0) = 0 + 0 + 0 = 0

u2 · u3 = (-3)(0) + (0)(7) + (2)(0) = 0 + 0 + 0 = 0

Since the dot product of every pair of vectors is zero, they are orthogonal.

   2.Linear Independence: We need to show that the vectors u1, u2, and u3 are linearly independent, meaning that no vector can be written as a linear combination of the other vectors.

We can determine linear independence by forming a matrix with the vectors as its columns and performing row operations to check if the matrix can be reduced to the identity matrix.

[A | I] = [u1 | u2 | u3 | I] =

[2 -3 0 | 1 0 0]

[0 0 7 | 0 1 0]

[3 2 0 | 0 0 1]

Performing row operations:

R3 - (3/2)R1 -> R3

R1 <-> R2

[1 0 0 | -3/2 1 0]

[0 1 0 | 0 1 0]

[0 0 7 | 0 0 1]

Since we can obtain the identity matrix on the left side, the vectors u1, u2, and u3 are linearly independent.

Therefore, the vectors u1 = (2, 0, 3), u2 = (-3, 0, 2), and u3 = (0, 7, 0) form an orthogonal basis for R^3.

(b) To write v = (1, 2, 3) as a linear combination of u1, u2, and u3, we need to find the coefficients x, y, and z such that:

v = xu1 + yu2 + z*u3

Substituting the given vectors and coefficients:

(1, 2, 3) = x(2, 0, 3) + y(-3, 0, 2) + z(0, 7, 0)

Simplifying the equation component-wise:

1 = 2x - 3y

2 = 7y

3 = 3x + 2y

From the second equation, we can solve for y:

y = 2/7

Substituting y into the first equation:

1 = 2x - 3(2/7)

1 = 2x - 6/7

7 = 14x - 6

14x = 13

x = 13/14

Substituting the found values of x and y into the third equation

3 = 3(13/14) + 2(2/7)

3 = 39/14 + 4/7

3 = 39/14 + 8/14

3 = 47/14

Therefore, we have determined the values of x, y, and z as follows:

x = 13/14

y = 2/7

z = 47/14

Thus, we can write the vector v = (1, 2, 3) as a linear combination of u1 = (2, 0, 3), u2 = (-3, 0, 2), and u3 = (0, 7, 0) as:

v = (13/14)u1 + (2/7)u2 + (47/14)u3

Therefore, v can be expressed as a linear combination of the given vectors.

Final Answer:Therefore,the linear combination of v = (13/14)u1 + (2/7)u2 + (47/14)u3  

To learn more about a vector as a linear combination of vectors using a system of equations from the given link

https://brainly.com/question/30758379

#SPJ4

The linear combination of v = (13/14)u1 + (2/7)u2 + (47/14)u3  

To express a vector as a linear combination of  vectors using a system of equations, we need to find the coefficients that multiply each given vector to obtain the desired vector. This can be done by setting up a system of equations, where each equation corresponds to the components of the vectors involved.

Body of the Solution:

(a) To show that the vectors u1 = (2, 0, 3), u2 = (-3, 0, 2), and u3 = (0, 7, 0) form an orthogonal basis for R^3, we need to demonstrate two conditions: orthogonality and linear independence.

Orthogonality: We need to show that each pair of vectors is orthogonal, meaning their dot product is zero.

u1 · u2 = (2)(-3) + (0)(0) + (3)(2) = -6 + 0 + 6 = 0

u1 · u3 = (2)(0) + (0)(7) + (3)(0) = 0 + 0 + 0 = 0

u2 · u3 = (-3)(0) + (0)(7) + (2)(0) = 0 + 0 + 0 = 0

Since the dot product of every pair of vectors is zero, they are orthogonal.

  2.Linear Independence: We need to show that the vectors u1, u2, and u3 are linearly independent, meaning that no vector can be written as a linear combination of the other vectors.

We can determine linear independence by forming a matrix with the vectors as its columns and performing row operations to check if the matrix can be reduced to the identity matrix.

[A | I] = [u1 | u2 | u3 | I] =

[2 -3 0 | 1 0 0]

[0 0 7 | 0 1 0]

[3 2 0 | 0 0 1]

Performing row operations:

R3 - (3/2)R1 -> R3

R1 <-> R2

[1 0 0 | -3/2 1 0]

[0 1 0 | 0 1 0]

[0 0 7 | 0 0 1]

Since we can obtain the identity matrix on the left side, the vectors u1, u2, and u3 are linearly independent.

Therefore, the vectors u1 = (2, 0, 3), u2 = (-3, 0, 2), and u3 = (0, 7, 0) form an orthogonal basis for R^3.

(b) To write v = (1, 2, 3) as a linear combination of u1, u2, and u3, we need to find the coefficients x, y, and z such that:

v = xu1 + yu2 + z*u3

Substituting the given vectors and coefficients:

(1, 2, 3) = x(2, 0, 3) + y(-3, 0, 2) + z(0, 7, 0)

Simplifying the equation component-wise:

1 = 2x - 3y

2 = 7y

3 = 3x + 2y

From the second equation, we can solve for y:

y = 2/7

Substituting y into the first equation:

1 = 2x - 3(2/7)

1 = 2x - 6/7

7 = 14x - 6

14x = 13

x = 13/14

Substituting the found values of x and y into the third equation

3 = 3(13/14) + 2(2/7)

3 = 39/14 + 4/7

3 = 39/14 + 8/14

3 = 47/14

Therefore, we have determined the values of x, y, and z as follows:

x = 13/14

y = 2/7

z = 47/14

Thus, we can write the vector v = (1, 2, 3) as a linear combination of u1 = (2, 0, 3), u2 = (-3, 0, 2), and u3 = (0, 7, 0) as:

v = (13/14)u1 + (2/7)u2 + (47/14)u3

Therefore, v can be expressed as a linear combination of the given vectors.

Therefore, the linear combination of v = (13/14)u1 + (2/7)u2 + (47/14)u3  

Learn more about linear combination

https://brainly.com/question/32297475

#SPJ4

Answer:3a ?

Step-by-step explanation:

Answer:

✈️?

Step-by-step explanation:

These aspects, among others, contribute to the overall aesthetic response to an Art Song. However, it's important to note that individual preferences and cultural backgrounds can greatly influence how a listener perceives and responds to the music. Each person's aesthetic response may be unique, and an Art Song can evoke a range of emotions and interpretations based on personal experiences and sensibilities.

Here, we have,

Instrumentation: The combination of one voice and piano in an Art Song creates an intimate and nuanced sonic palette. The piano provides harmonic support and textures while allowing the voice to take the lead. The delicate interplay between the voice and piano can evoke a sense of intimacy and draw the listener into the music.

Style of Singing: Art Songs are often characterized by a lyrical and expressive style of singing. The singer's ability to convey emotions, communicate the meaning of the text, and deliver a captivating performance can leave a lasting impression. The nuances in phrasing, dynamics, and vocal technique contribute to the overall aesthetic response.

Language of Text: The language used in an Art Song can evoke different aesthetic responses depending on the listener's familiarity with the language and cultural background. The choice of language may enhance the poetic quality of the lyrics, evoke specific imagery or cultural associations, and deepen the emotional connection to the music.

Word Painting: Art Song composers frequently use word painting techniques to musically illustrate or depict specific words or phrases in the text. These musical gestures can include melodic contour, rhythmic patterns, dynamic contrasts, and harmonic choices. Word painting enhances the listener's understanding and emotional engagement by creating vivid musical images and reinforcing the meaning of the text.

Emotional Range: Art Songs often explore a wide range of emotions, from melancholy and introspection to joy and passion. The music's ability to evoke and express these emotions can elicit a profound aesthetic response in listeners. The shifts in mood, dynamics, and melodic lines can create an emotional journey that resonates with the listener's own experiences and feelings.

Musical Storytelling: Art Songs can tell stories or convey narratives through the integration of music and text. The composer's use of melodies, harmonies, and rhythmic patterns can depict characters, events, and landscapes, enabling the listener to engage with the story being told. The unfolding narrative and the music's ability to convey the story's essence can captivate and emotionally move the listener.

These aspects, among others, contribute to the overall aesthetic response to an Art Song. However, it's important to note that individual preferences and cultural backgrounds can greatly influence how a listener perceives and responds to the music. Each person's aesthetic response may be unique, and an Art Song can evoke a range of emotions and interpretations based on personal experiences and sensibilities.

learn more on Art Songs

https://brainly.com/question/30269231

#SPJ4

Answer:

1: (4,1) x=4 y=1

2: (1,10) x=1 y=10

3: (-25/4,5) x= -25/4 y=5

Step-by-step explanation:

It would be a lot to explain all three so if you really need it just ask

An equivalent function can be written as \(G(t) = 1920 \times 2^t\)

What is equivalent fraction?

Fractions that represent the same amount or portion of a whole, but have different denominators and numerators is called Equivalent fractions.

They are equal in value, but may have different forms.

We can multiply or divide both the numerator and the denominator of a fraction by the same number for find equivalent fractions.

Some examples are 2/4 is equivalent to 1/2 because if you divide both the numerator and the denominator of 2/4 by 2, you get 1/2.

Starting from the given function, we can simplify it as

\(G(t) = 960(2^t + 1) \\ G(t) = 960 \times 2^t \times 2^1 \\ G(t) = 1920 \times 2^t\)

We can see that this function is of the form

\(G(t) = ab^t\)

where a = 1920 and b = 2.

Therefore, an equivalent function can be written as \(G(t) = 1920 \times 2^t\)

Learn more about equivalent fraction here,

https://brainly.com/question/17912

#SPJ1

Answer:

12

Step-by-step explanation:

2DE = BC

8n = 6n + 6

2n = 6

n = 3

4n = 4(3) = 12

Answer:

slope = (3/2)

Step-by-step explanation:

point 1: (-1, 1); Point 2: (1, 4)

            (x₁, y₁)              (x₂, y₂)

y₂ - y₁         4 - 1            3            3

----------- = ------------ = --------- = -------

x₂ - x₁          1 - (-1)       1 + 1          2

I hope this helps!

Answer:

160

Step-by-step explanation:

Just divide 1920 into 12

Answer:

y = - \(\frac{5}{2}\) x - 1

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = \(\frac{y_{2}-y_{1} }{x_{2}-x_{1} }\)

with (x₁, y₁ ) = (- 4, 9) and (x₂, y₂ ) = (2, - 6) ← 2 ordered pairs from the table

m = \(\frac{-6-9}{2-(-4)}\) = \(\frac{-15}{2+4}\) = \(\frac{-15}{6}\) = - \(\frac{5}{2}\)

the line crosses the y- axis at point (0, - 1 ), ordered pair from table, then

c = - 1

y = - \(\frac{5}{2}\) x - 1 ← equation of line

Answer:

There are 39 seats I think.

Step-by-step explanation:

548 divided by 14 is a decimal but rounded it to the nearest full number.

The price of the table yesterday is $925

The sale price of the table today = $592

Let yesterday's sale price be x

Discount = 36%

The percentage sale price today = 100% - 36%

The percentage sale price today = 64%

Today's sale price = 64% of yesterday's sale price

592 = (64/100)x

592 = 0.64x

x = 592/0.64

x = $925

The price of the table yesterday is $925

Using the weighted mean, the coordinates of P are given as follows:

The weighted mean is given by the sum of all elements in a data-set multiplied by it's weight, divided by the sum of the weights.

In this problem, we have these following points:

Applying the weighted mean, the coordinates of P are given as follows:

In which xA and yA are the coordinates x and y of point A, and xB and yB are the coordinates x and y of point B.

More can be learned about the weighted mean at https://brainly.com/question/24398353

#SPJ1

using standard z value, 20% of sandwiches are less than 10.58 inches.

In the given question,

Subway sells foot long sandwiches that have a mean of 11 inches and a standard deviation of 0.5 inches.

20% of sandwiches are less than...............inches.

The cumulative standardized normal distribution table indicates a z value of -0.84 for 20%.

From the question we know that

Mean(μ) = 11 inches

Standard Deviation(σ) = 0.5 inches

We have to find less than of 20% of sandwiches for z value of -0.84

P(ƶ<z) = 20%

We can write it as

P(ƶ<-0.84) = 20/100

P(ƶ<-0.84) = 0.20

Since z = -0.84

X-μ/σ = -0.84

Now putting the value

X-11/0.5 = -0.84

Multiply by 0.5 on both side

X-11/0.5 *0.5= -0.84*0.5

X-11= -0.42

Add 11 on both side

X-11+11= -0.42+11

X = 10.58

Hence, 20% of sandwiches are less than 10.58 inches.

To learn more about standard z value link is here

https://brainly.com/question/27315585

#SPJ4

Right question is here

Subway sells foot long sandwiches that have a mean of 11 inches and a standard deviation of 0.5 inches. 20% of sandwiches are less than ................. inches. (The cumulative standardized normal distribution table indicates a z value of -.84 for 20%)

(a) 11.500

(b) 11.42

(c) 10.58

(d) cannot be determined from the information given

Susan earns $9.50 per hour. This is found by dividing her earnings by the number of hours worked for each corresponding row in the table.

To find how much money Susan earned per hour, we need to divide the total earnings by the total number of hours worked. For finding the Total earnings we need to add the money earned in every hour,

Total earnings = $47.50 + $66.50 + $85.50 + $104.50 = $304

Total hours worked = 5 + 7 + 9 + 11 = 32

Money earned per hour = Total earnings / Total hours worked

= $304 / 32

= $9.50

Therefore, Susan earned money of $9.50 per hour.

To know more about Money earned:

https://brainly.com/question/13510603

#SPJ4

Two lines are perpendicular if:

For the line:

We can see that its slope is 2/3, so:

Let:

m1 = 2/3

Therefore:

Now, we have the slope and a point P(6,6), let's use the slope point equation:

Answer:

A 1.8 × 10⁸

D 8.0 × 10⁶

B 1.2 × 10⁶

C 2.4 × 10²

Step-by-step explanation:

the scientific notation is nothing else than 10 to the power of an exponent multiplied by the first number.

that exponent of 10 makes the main decision about the size. the higher the bigger.

if the exponents are equal, then the first number decides.

10⁸ is the largest number compared to the others.

then 10⁶, then 10².

but beware of tricksters among teachers and testers.

what I just explained works without any additional steps only if the scientific notation is truly followed.

that means the first number has exactly 1 position left of the decimal point, and it is not 0.

but anybody could write something like

3126.9 × 10⁴

it is not scientific notation, but it is a valid number in general.

so, is

3126.9 × 10⁴

smaller or larger than A, B, C, D ?

we get the answer by converting the general number into scientific notation, and THEN we compare :

3126.9 × 10⁴ = (3.1269 × 10³) × 10⁴ = 3.1269 × 10⁷

aha !

it is smaller than A but larger than D and B.

and yes, of course, larger than C.

Answer:

10 times 4 ten thousands = 40 ten thousands

Step-by-step explanation:

Answer:

25%

Step-by-step explanation:

As we are measuring the area of a two-dimensional surface in square units, the unit for the bean field's area is square feet.

A two-dimensional object, such as a rectangle, square or triangle, has a perimeter that may be calculated as the distance encircling the edge. The lengths of all the shape's sides are added up to determine it.\(P = 2(l + w)\), where P is the perimeter, l is the length, and w is the width of the rectangle, illustrates how the perimeter of a rectangle is determined by adding the lengths of each of its four sides. In building, architecture, and engineering, perimeter is frequently used to calculate the amount of material required for a structure's edges or to establish the length of a fence or boundary.

given,

By multiplying the first two elements in this expression using the distributive property, we may simplify it to:

\(V=(4x^{2}+8x+4)*(x+3)\)

Using the distributive property to expand this statement, we obtain:

\(V=4x^{3}+20x^{2}+32x+12\)

Hence, the polynomial expression for the bean plants' volume assuming they grow to a height of \((x + 2)\) feet is \(4x3 + 20x2 + 32x + 12.\)

We must enter \(x = 3\) in the calculation for the area of the rectangle in order to get the size of the bean field when \(x = 3\). As the area of a rectangle is equal to its length times its breadth, the area of the bean field is:

\(A = (2(3) + 2) * (3(3) + 2) = 20\)

The bean field therefore has a \(20\) square foot space.

As we are measuring the area of a two-dimensional surface in square units, the unit for the bean field's area is square feet.

To know more about perimeter visit:

brainly.com/question/6465134

#SPJ1

The correct option is option (d) . That's a zero order relationship means that there is no test factors have been controlled.

In statistics, there are many types of correlations that can be performed to determine the relationship between two variables. These types of correlations include Pearson's correlation, Spearman's correlation, point biserial correlation, and Kendall's correlation. You may also have come across the terms zero-order, partial, and partial in relation to correlation. These terms refer to correlations involving three or more variables.

A zero-order relationship means that the zero-order correlation simply refers to the correlation between his two variables (i.e. independent and dependent variables) and does not control for the possible influence of the other variable.

To learn more about Zero-Order Correlation, refer:

https://brainly.com/question/11539028

#SPJ4

Answer: 19

Step-by-step explanation:

Answer: it is 71,because 180-109=71

Step-by-step explanation:

Answer:

a) Red = 0.2 Blue= 0.4 and White =0.4

b) 90 counters in total

Step-by-step explanation:

a)Firstly we know that the probability of mutually exclusive events is equal to 1 and the question states that the probability of blue and white counter is the same so we subtract 1-0.2 = 0.8 and we divide that by 2 as a result 0.4 .

b) P= Favourable outcomes / total number of outcomes

P(red) = 0.2

total number of red counters = 18

0.2 = 18/t cross multiplying have 0.2t= 18

Dividing throughout by 0.2 having 90

The reflection of the given points across the x-axis is:
(-2 (3/4), -1), (1 1/4. 1/2), (-4, 2 1/2), and (3/4,-3).

A reflection is a mapping from an isometric hyperplane with a set of fixed points acting as the reflection's axis or plane from a Euclidean space to itself. A figure's reflection creates its mirror image of that figure in the plane or axis of the reflection.

The given ordered pairs are (-2 (3/4), 1), (1 1/4. -1/2), (-4, -2 1/2), and (3/4,3).

The reflection of a point (x,y) across the x-axis is (x, -y).

Therefore, the reflection of the given points across the x-axis is:
(-2 (3/4), -1), (1 1/4. 1/2), (-4, 2 1/2), and (3/4,-3).

Learn more about the reflection:
https://brainly.com/question/15487308

#SPJ1

Answer:

-9/14 :)))))))))))))))))))

If you are worried about your future in math and want to improve your mathematical abilities, the best way to do so is to practice regularly and seek out additional resources to help you understand the material.

Here are some specific things you can do:

Practice problems: Work through a variety of different types of problems to build your skills and confidence.

Find a study group: Join a study group with other students who are also taking the same math class. This can be a great way to get extra help and support.

Use online resources: There are many online resources, such as videos and tutorials, that can help you understand mathematical concepts.

Seek extra help: Don't be afraid to ask your teacher or a tutor for extra help if you are struggling with a particular topic.

Stay motivated: Keep in mind the importance of math in many fields and how it will be useful in future, and set realistic and challenging goals for yourself.

Make connections between different topics: Try to understand how different mathematical concepts are related and how they can be applied in real-world situations.

Remember, learning math takes time and practice, so be patient and persistent.

Learn more about mathematics on:

https://brainly.com/question/27097057

#SPJ1

Answer:

10 inches/hour

Step-by-step explanation:

The tip moves clockwise in a circle of radius 10 inches.  The linear velocity is v = rω, where r is the radius and ω the angular velocity.  

This comes out to v = (10 inches)(1 revolution per hour) = 10 inches/hour

Answer:

x²^41

Step-by-step explanation:

x²^41 combine the factors only

Answer:

y = 4/3x - 4 , you just have to place the plots on a graph and find the slope: 4/3x and the y-intercept is obviously -4.

Step-by-step explanation:

Answer:

Step-by-step explanation:

Given:

Number of half-life periods:

Equation for remaining sample:

Answer:

a, b, and c are represented by the graph.

Step-by-step explanation:

i hope this helps :)