Let f: X → R be a linear function, where X is a topological vector space. (a) Suppose that f is bounded above on a neighborhood V of the origin. That means to 7>0 such that f(x) ≤ y for all x € V. Prove that there exists a neighborhood W of the origin such that f(x)| ≤ y for all x € W. (b) Suppose that f is bounded above on a neighborhood V of the origin. Prove that f is co (c) Prove that if f is bounded above on a set 2 with int(2) Ø, then f is continuous.
Answers
(a) To prove that there exists a neighborhood W of the origin such that f(x) ≤ y for all x ∈ W, given that f is bounded above on a neighborhood V of the origin, we can use the linearity of f.
Since f is a linear function, it satisfies the following properties:
f(0) = 0
f(rx) = rf(x) for any scalar r and vector x
f(x + y) = f(x) + f(y) for any vectors x and y
Given that f is bounded above on V, there exists a positive number M such that f(x) ≤ M for all x ∈ V. Now, let's consider the neighborhood W defined as follows:
W = {x ∈ X | ||x|| < M}
We claim that for any x ∈ W, f(x) ≤ y.
Let x ∈ W. Since x is in the neighborhood W, we have ||x|| < M. By linearity, we can express x as x = rx' for some scalar r and vector x' with ||x'|| = 1.
Now, consider f(x):
f(x) = f(rx') = rf(x')
Since ||x'|| = 1, we have ||rx'|| = |r| ||x'|| = |r|.
Therefore, ||rx'|| < M implies |r| < M.
Using the fact that f is bounded above on V, we have f(x') ≤ M.
Combining these results, we get:
|f(x)| = |rf(x')| = |r| |f(x')| ≤ M
Since this inequality holds for any x ∈ W and |r| < M, we have shown that f(x) ≤ y for all x ∈ W, where W is a neighborhood of the origin.
(b) To prove that f is continuous, we can show that f is bounded above on any compact set in X. Let K be a compact set in X.
Since K is compact, it is also closed and bounded. By the linearity of f, we have:
f(K) = {f(x) | x ∈ K}
Since K is bounded, there exists a positive number M such that ||x|| ≤ M for all x ∈ K. By the linearity of f, we have:
f(K) = {f(x) | x ∈ K} ⊆ {f(x) | ||x|| ≤ M}
Thus, f(K) is bounded above by M.
By the previous result in part (a), if f is bounded above on a neighborhood of the origin, then it is bounded above on any neighborhood of the origin. Therefore, f is bounded above on the neighborhood V of the origin.
Since K is compact, it can be covered by finitely many neighborhoods of the origin, say V1, V2, ..., Vk. Thus, f is bounded above on each Vi, i = 1, 2, ..., k.
Now, consider the open cover {V1, V2, ..., Vk} of K. By compactness, there exists a finite subcover {V1, V2, ..., Vm}. Therefore, f is bounded above on K.
Since f is bounded above on any compact set K, it follows that f is continuous.
(c) The previous part (b) already proves that if f is bounded above on any compact set, it is continuous. Therefore, if f is bounded above on a set 2 with int(2) ≠ Ø (i.e., the interior of 2 is not empty), then f is continuous.
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Related Questions
A washing machine in a laundromat breaks down an average of three times per
month. Using the Poisson probability distribution formula, find the probability that
during the next month this machine will have
a) exactly two breakdowns
b) at most one breakdown.
Answers
a) the probability of exactly two breakdowns in the next month is approximately 0.224 or 22.4%
b) the probability of at most one breakdown in the next month is approximately 0.199 or 19.9%.
To find the probability of the number of breakdowns in a month for the washing machine using the Poisson probability distribution, we need to know the average rate of breakdowns per month, which is given as three breakdowns.
The Poisson probability distribution formula is given by:
P(x; λ) = (\(e^{(-\lambda )\)* λ^x) / x!
Where:
P(x; λ) is the probability of x events occurring in a given interval,
λ is the average rate of events occurring in that interval,
e is the base of the natural logarithm, and
x! represents the factorial of x.
a) To find the probability of exactly two breakdowns, we substitute x = 2 and λ = 3 into the formula:
P(2; 3) = (\(e^{(-3)} * 3^2\)) / 2!
Calculating this, we get:
P(2; 3) ≈ 0.224
Therefore, the probability of exactly two breakdowns in the next month is approximately 0.224 or 22.4%.
b) To find the probability of at most one breakdown, we sum the probabilities of zero breakdowns and one breakdown:
P(0; 3) + P(1; 3)
Substituting x = 0 and λ = 3 into the formula for the first term:
P(0; 3) = (\(e^{(-3)} * 3^0)\) / 0! = \(e^{(-3)\)
Substituting x = 1 and λ = 3 into the formula for the second term:
P(1; 3) = (\(e^{(-3)} * 3^1\)) / 1! =\(3e^{(-3)\)
Calculating this sum, we get:
P(0; 3) + P(1; 3) = \(e^{(-3)} + 3e^{(-3)}\)
Approximating this value, we find:
P(0; 3) + P(1; 3) ≈ 0.199
Therefore, the probability of at most one breakdown in the next month is approximately 0.199 or 19.9%.
Using the Poisson probability distribution, we can determine the likelihood of different numbers of breakdowns occurring in a month for the washing machine in the laundromat.
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help plssss!!
Which answer describes the function f(x)=2x3−x2 ?
odd
even
neither
Answers
the answer is neither
Which of the following functions f has a removablediscontinuity at a? If the discontinuity is removable,find a function g that agrees with f forxa and is continuous at a. (If thediscontinuity is not removable, enter NONE.)
\(a\) \ f(x) = (x^4 - 1)/(x-1), a = 1
a) The discontinuity isremovable.
b) The discontinuity is notremovable.
Answers
a) The discontinuity is removable.
To find the limit of f(x) as x approaches 1, we can factor the numerator as a difference of squares:
f(x) = [(x^2 + 1)(x^2 - 1)] / (x - 1) = [(x^2 + 1)(x + 1)(x - 1)] / (x - 1) = (x^2 + 1)(x + 1)
Notice that the (x - 1) terms cancel out, leaving us with f(x) = (x^2 + 1)(x + 1). Therefore, we can redefine f(x) at x = 1 by setting g(x) = f(x) = (x^2 + 1)(x + 1), which is continuous at x = 1.
what is factor?
In mathematics, a factor refers to a number or algebraic expression that can be multiplied by another number or expression to produce a product. For example, in the expression 5x + 10, the factors are 5 and (x + 2). When multiplied together, they produce the product 5(x + 2). In general, factors can be used to simplify expressions, solve equations, and factorize polynomials.
what is number?
A number is a mathematical concept used to count, measure, and label. It can be used to represent quantity, order, position, and more. Numbers can be positive, negative, or zero, and they can be whole or fractional. The most commonly used types of numbers are integers, rational numbers, real numbers, and complex numbers.
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setting up squares of known dimension over the entire pattern describes what method of documentation?
Answers
The method of documentation that describes the setting up of squares of known dimension over the entire pattern is known as grid method.
Grid method is a method of documentation that involves the setting up of squares of known dimensions over the entire pattern to document a site. This method is commonly used in archaeological surveys and historical preservation.
Grids are used to help maintain a sense of orientation and scale in the documentation process, and they provide a reference point for mapping and recording data. Grids can be set up on maps or site plans, as well as on the ground using physical markers such as stakes or string.Grids can also be used to help facilitate excavation by allowing archaeologists to isolate specific areas of a site and keep track of their findings in a consistent and organized manner.Therefore, grid method is a simple and effective way of recording data, and it is still widely used in the field of archaeology today.
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Quadrilateral ABCD is inscribed in a circle where BD is a diameter of the circle and m/ADC = 62°. D m/DAB Find the measures of the other three angles of the quadrilateral. m/ABC= A m/BCD
Answers
The value of angle DAB is 90 degrees.
The value of angle ABC is 118 degrees.
The value of angle BCD is 90 degrees.
What is the measure of the missing angles of the quadrilateral?The measure of the missing angles of the quadrilateral inscribed in a circle is calculated by applying circle theorem as follows;
If line BD is the diameter, then we will have the following;
angle BCD = 90 degrees
angle DAB = 90 degrees
Now, the value of angle ABC is calculated as follows;
angle ABC = 180 - angle ADC ( opposite angles of a cyclic quadrilateral are supplementary).
angle ABC = 180 - 62⁰
angle ABC = 118⁰
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Find a counterexample for this statement.
A quadrilateral has four congruent sides.
Answers
Answer:
A Rhombus
Step-by-step explanation:
it's a quadrilateral, but its sides aren't congruent
we want a cu-30% zn brass plate originally 1.54 inch thick to have an elongation greater than 36 %. what is the smallest final thicknesses that can be obtained?
Answers
The smallest final thickness that can be obtained for the Cu- 30% Zn brass plate is approximately 2.0944 inches.
To calculate the smallest final thickness that can be obtained for a Cu- 30% Zn brass plate with an elongation greater than 36%, we need to consider the relationship between elongation and thickness.
Elongation is a measure of the material's ability to stretch or deform without breaking. It is typically expressed as a percentage increase in length compared to the original length. In this case, the elongation needs to be greater than 36%.
To calculate the smallest final thickness, we can use the formula:
Final thickness = Original thickness * (1 + Elongation/100)
Let's plug in the given values:
Original thickness = 1.54 inch
Elongation = 36%
Final thickness = 1.54 inch * (1 + 36%/100)
Final thickness = 1.54 inch * (1 + 0.36)
Final thickness = 1.54 inch * 1.36
Final thickness = 2.0944 inch (rounded to four decimal places)
Therefore, the smallest final thickness that can be obtained for the Cu-30% Zn brass plate is approximately 2.0944 inches.
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A line contains the points (8,9) and (-12, -7). Using point-slope form, write the equation of the line that is parallel to the given line
and that passes through (-5, -15).
A
4
B
3-15= ( 9
» +15= 3(+9
3+5= 3(+15)
»+15= 3(x+5)
C
D
Answers
Answer:
(y+15) = 4/5(x+5)
Step-by-step explanation:
slope = (-7 - 9)/(-12 - 8) = -16/-20 = 4/5
(y+15) = 4/5(x+5)
2x - 4y = b₁ -x + 5y = b₂ Consider the above 2*2 linear system and answer the following questions: 1) If b = [], draw the row picture. [1] draw the I column picture. -1 1 0 A = 0 -1 1 0 0 For the above matrix A, answer the following questions: (a) If b=(-1, -4, -6), discuss the solution for Ax=b . (b) If b=(0, 0, 0), discuss the solution for Ax=b . 2) If b =
Answers
For the given matrix A and different values of b, (a) with b = (-1, -4, -6) has a unique solution, while (b) with b = (0, 0, 0) has infinitely many solutions forming a line.
In the given linear system of equations, 2x - 4y = b₁ and -x + 5y = b₂, where b₁ and b₂ are constants, we are asked to analyze the solutions for different values of b.
(a) If b = [-1, -4, -6]:
To find the solution for Ax = b, we perform row operations on the augmented matrix [A | b]. Row-reducing it leads to [I | x] = [[1, 0, -2], [0, 1, -2], [0, 0, 0]]. Here, x = [-2, -2, 0], indicating that the system is consistent with a unique solution. The solution is x = -2, y = -2.
(b) If b = [0, 0, 0]:
Again, row-reducing the augmented matrix gives [I | x] = [[1, 0, 0], [0, 1, 0], [0, 0, 0]]. Here, x can take any values since the third row is all zeros. This implies the system has infinitely many solutions, forming a line in 3D space.
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What is the right translation of these expressions and equations? (with solution)
1. 7 - 2m
2. 3( m + 2) = 15
3. 5m - m(2 - m)
Answers
Answer:
7 - 2m can be translated to "7 minus two times m" or "the difference between 7 and twice m".
3(m + 2) = 15 can be translated to "three times the sum of m and 2 is equal to 15" or "the product of 3 and the sum of m and 2 is 15".
To solve the equation, we can start by distributing the 3 on the left side:
3(m + 2) = 15
3m + 6 = 15
Then, we can subtract 6 from both sides:
3m + 6 - 6 = 15 - 6
3m = 9
Finally, we can divide both sides by 3:
3m/3 = 9/3
m = 3
Therefore, the solution to the equation 3(m + 2) = 15 is m = 3.
5m - m(2 - m) can be translated to "5m minus the product of m and the difference between 2 and m" or "the difference between 5m and m times the quantity 2 minus m".
To simplify the expression, we can use the distributive property to expand the second term:
5m - m(2 - m) = 5m - 2m + m^2 = m^2 + 3m
Therefore, the simplified expression is m^2 + 3m.
Parisian sewer #1 contains 9 fewer gigantic rats—rats as big as cats, some might say—than parisian sewer #2. parisian sewer #2 contains 5 fewer gigantic rats than parisian sewer #3. if there are 56 gigantic rats in all three sewers combined, how many rats are in sewer #1?
Answers
There are different kinds of math problem. There will be 11 rats in sewer #1.
What are word problem?The term word problems is known to be problems that are associated with a story, math, etc. They are known to often vary in terms of technicality.
Lets take
sewer #1 = a
sewer #2 = b
sewer #3 = c
Note that A=B-9
So then you would have:
A=B-9
B=C- 5
A+B+C=56
Then you have to do a substitution so as to find C:
(B- 9) + (C-5) + C = 56
{ (C- 5)-9} + (C-5) + C = 56
3C - 19 = 56
3C = 75
B = C- 5
B = 25 - 5
Therefore, B = 20
A = B - 9
= 25 - 9
=11
Therefore, there are are 11 rats in sewer #1
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There are 55 students in mr.fishers p.e class. There are 30 6th graders and 25 7th graders in the class. What is the ratio of 6th graders to 7th graders in nr.fishers class
Answers
Answer:
6:5
Step-by-step explanation:
It's 6:5 trust homie
What decimal is equivalent to 15%
Answers
Answer:
0.15 is the right answer
Answer:
0.15
Step-by-step explanation:
15% is equivalent to the decimal 0.15. Notice that dividing by 100 moves the decimal point two places to the left.
if m<4=35 degrees, find m<2 and m<3
Answers
35, ∠2 and ∠4 are alternate interior angles, so m∠2 = m∠4.
What is alternate interior angles?
Two angles that are on the interior of and, but on different sides of the transversal, are said to be alternate interior angles. The alternate interior angles theorem states that the alternate interior angles must be congruent if two parallel lines are cut by a transversal.Given
m∠4=35
From the image given, angle 2 and angle 4 lie on opposite side of the line that intercepts the two parallel lines, AB and CD. Angle 2 and angle 4 both lie within the parallel lines.
Therefore, ∠2 and ∠4 are alternate interior angles.
Thus,
m ∠2 = m∠4 (alternate interior angles theorem)
since m∠4 = 35°
therefore m∠2 = 35°
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17. What is the value of C so that x^2+15x + c is a perfect square trinomial?
A. 30
B. 15/2
C. 15/4
D. 225/4
Answers
Answer:
D. 225/4
Step-by-step explanation:
To complete the square (a technique that can be applied to many problems!), take half the coefficient of x and square the result.
The coefficient of x is 15. Half of 15 is 15/2. Square 15/2 to get 225/4.
Check:
\(x^2+15x+\frac{225}{4}=\left(x+\frac{15}{2}\right)\left(x+\frac{15}{2}\right)=\left(x+\frac{15}{2}\right)^2\)
Answer:
255/4
Step-by-step explanation:
write an expression for the area of the following quadrilateral with the fewest terms. all sides are congruent. ALSO PLEASE SHOW ME HOW YOU GOT IT
Answers
Explanation: All sides are congruent which means they are equal. This means that you can just multiply the amount of sides to the length of one side.
Hope this helps! :)
Word problem with 9.5*(8+12.5)
Answers
(12.5 x 9.5)= 118.75
76+118.75= 194.75
Answer:
88.5
Step-by-step explanation:
First you multiple 9.5 and 8 and you should get 76.0
last you add 12.5 and 76.0 and you should get 88.5
and make sure you line up the decimals when adding.
hope this helps
In the data set below, what is the mean absolute deviation?
8 4 2 3 6
If the answer is a decimal, round it to the nearest tenth.
mean absolute deviation (MAD):
Answers
The mean absolute deviation of the given data set is 1.6 (rounded to the nearest tenth).
We have,
In statistics, the mean (also known as the arithmetic mean or average) is a measure of central tendency that represents the sum of a set of numbers divided by the total number of numbers in the set.
To find the mean absolute deviation (MAD), we need to first calculate the mean of the given data set:
mean = (4 + 5 + 7 + 9 + 8) / 5 = 6.6
Next, we calculate the deviation of each data point from the mean:
|4 - 6.6| = 2.6
|5 - 6.6| = 1.6
|7 - 6.6| = 0.4
|9 - 6.6| = 2.4
|8 - 6.6| = 1.4
Then we find the average of these deviations, which gives us the mean absolute deviation:
MAD = (2.6 + 1.6 + 0.4 + 2.4 + 1.4) / 5 = 1.6
Therefore, the mean absolute deviation of the given data set is 1.6 (rounded to the nearest tenth).
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W is not a subspace of the vector space. Verify this by giving a specific example that violates the test for a vector subspace (Theorem 4.5).
W is the set of all vectors in R3 whose components are nonnegative.
Answers
The resulting vector (0, 1, 0) is not in W because it has a negative component. This violation of closure under vector addition shows that W is not a subspace of the vector space R3.
To show that W is not a subspace of the vector space, we need to find a specific example that violates the test for a vector subspace (Theorem 4.5).
Theorem 4.5 states that for a set to be a subspace, it must satisfy three conditions:
1. The set contains the zero vector.
2. The set is closed under vector addition.
3. The set is closed under scalar multiplication.
Let's consider the second condition. To violate it, we need to find two vectors in W whose sum is not in W.
Let u = (1, 2, 3) and v = (4, 5, 6). Both u and v have nonnegative components, so they belong to W.
However, their sum u + v = (5, 7, 9) does not have nonnegative components, so it does not belong to W. Therefore, W is not closed under vector addition and is not a subspace of the vector space.
In summary, we have shown that W is not a subspace of the vector space by providing a specific example that violates the test for a vector subspace.
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A rectangular roller skating rink has no boundaries. If you were to take a 300-foot length of rope to create boundaries around the rink, what dimensions should be used to surround the rink to maximize area?
Answers
Firstly, let's assume that the length of the rectangular roller skating rink is L and the width is W. Also, let's assume that the length of the rope is 300 feet.
Now, we need to find the dimensions that will maximize the area of the rectangular rink. We know that the perimeter of the rink is equal to the length of the rope. Therefore, we can write an equation:
2L + 2W = 300
Simplifying this equation, we get:
L + W = 150
We need to maximize the area of the rink, which is given by:
A = LW
We can use the equation L + W = 150 to express one of the variables in terms of the other. For example, we can write:
L = 150 - W
Substituting this expression into the equation for the area, we get:
A = (150 - W)W
Simplifying this equation, we get:
A = 150W - W^2
To find the maximum area, we need to find the value of W that maximizes this equation. We can do this by taking the derivative of the equation with respect to W and setting it equal to zero:
dA/dW = 150 - 2W = 0
Solving for W, we get:
W = 75
Substituting this value of W into the equation for the perimeter, we get:
L = 75
Therefore, the dimensions that should be used to surround the rink to maximize area are 75 feet by 75 feet.
In conclusion, the maximum area of the rink can be achieved by using a length of rope of 300 feet to create boundaries around a rectangular rink with dimensions of 75 feet by 75 feet.
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If c= 2m + d, which expression represents m?
Answers
Answer:
M+c/2-d/2
Step-by-step explanation:
A local hamburger shop sold a combined total of 628 hamburgers and cheeseburgers on Thursday. There were 72 fewer cheeseburgers sold than hamburgers
How many hamburgers were sold on Thursday?
Answers
Answer:
278 cheeseburgers and 350 hamburgers sold on Thursday
Step-by-step explanation:
To confirm that these combined equal 628 we can add them together using the equation 278 + 350. Once confirm that these both equal 628 when combined, we can find the difference between the 350 hamburgers and 278 cheeseburgers by subtracting one from the other using the equation, 350 - 278. When you do that, you will get a difference of 72 which means that there were 72 fewer cheeseburgers sold than hamburgers.
(sorry if this is confusing, I'm not good at explaining things)
Estimate the sum to the nearest tenth: (-2.678) 4.5 (-0.68) what is the actual sum?
Answers
The actual sum is 1.1.
Sum of the given numbers (-2.678), 4.5, (-0.68)
The sum can be defined as the result or answer after adding two or more numbers or terms. Thus, the sum is a way of putting things together.
(-2.678) + 4.5 + (-0.68)
= -2.678 + 4.5 - 0.68
= 4.5 - 3.358
= 1.142
Nearest tenth:
since .1 is in the tenth place and 42 isn't greater than 50, the answer would be 1.1.
The actual sum is 1.1.
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Finding missing angles
Answers
Answer:
x = 55
Step-by-step explanation:
We know that an angle showing a box is 90 degrees.
x+ 35 = 90
x = 90-35
x = 55
The missing angle labeled x is 55 degrees.
If a runner's position in feet is given by LaTeX: s(t) = 10 + 2t^2, then the runner's average speed between LaTeX: t = 1 second and LaTeX: t = 3 seconds is …
Answers
The runner's average speed between `t = 1` second and `t = 3` seconds is `8 feet/second.`
Given, the runner's position in feet is given by `s(t) = 10 + 2t²`.
The average speed is given by the formula:
Average speed = `(distance traveled) / (time taken)`
The distance traveled by the runner is equal to the displacement (change in position) between `t = 1` second and `t = 3` seconds.
The displacement between `t = 1` second and `t = 3` seconds is given by the difference between their positions:
Displacement = `s(3) - s(1)`
Putting `t = 3` in the given equation, we get:
s(3) = 10 + 2(3)² = 28Feet
Putting `t = 1` in the given equation, we get:s(1) = 10 + 2(1)² = 12Feet
Therefore, the displacement between `t = 1` second and `t = 3` seconds is:
Displacement = `s(3) - s(1)`
= 28 - 12
= 16 Feet
The time taken is
`t = 3 - 1
= 2` seconds
Therefore, the average speed is given by:
Average speed = `distance/time`
= `16 / 2`
= 8 feet/second
Hence, the runner's average speed between `t = 1` second and `t = 3` seconds is `8 feet/second.`
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if s.p = rs 450,s.p with vat =rs 495,find vat percent
Answers
The vat percent is 10%.
Given that s.p.=Rs 450 and s.p with vat =Rs 495.
We want to find Amount of VAT that is how much value added tax is imposed in the selling price.
firstly, we will find Amount of VAT by taking Selling Price (S.P) and subtract it from Selling Price with VAT.
Amount of VAT = (S.P + VAT )- S.P
Amount of VAT = Rs 495 - Rs 450
Amount if VAT = Rs 45
Now, we will find the vat percent by dividing amount of VAT with selling price and multiply with 100, we get
VAT Percent = (Amount of VAT /Selling Price)×100
VAT Percent =(45/450)×100
VAT Percent=10%
Hence, the VAT percentage is 10% when the selling price is Rs. 450 with the VAT selling price Rs 495.
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In the diagram below, AB is parallel to cd what is the value of x?
Answers
Answer:
45 degrees
Step-by-step explanation:
corresponding angles have are equal
28-32. Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value ofn. Then use Theorem 10. 18 to find an upper bound for the error S-Sn in using the nth partial sum Sn to estimate the value of the series S n=3 k-1 1 THEOREM 10. 18 Remainder in Alternating Series Let -1a be a convergent aiternating series with terms that are nonincreasing in magnitude. Let R-S-S, be the remainder in approximating the k-1 33-38. Remainders in alternating series Determine how mamy tems of the following convergent series must be summed to be sure that the remainder is less than 104 i magnitude Although you do not need it, the exact value of the series is ghven tn each case 34. - e k-0 k!
Answers
Theorem 10.18 states that the remainder R-Sn is bounded by the absolute value of the next term in the series, which is also the absolute value of the (n+1)th term. To determine how many terms of a given convergent series must be summed to ensure that the remainder is less than
\(10 { }^{ - 4} \)in magnitude.
We are given an alternating series, and we need to estimate the error in using the nth partial sum to approximate the sum of the series. This is a useful tool for estimating the error in approximating the sum of a series.
To apply Theorem 10.18, we need to evaluate the nth partial sum for the given value of n and find the absolute value of the (n+1)th term. We can use these values to estimate the error in approximating the sum of the series.
This is a common question in numerical analysis and involves estimating the error in approximating the sum of a series and then choosing the number of terms needed to achieve a desired level of accuracy.
These problems involve using techniques from calculus and numerical analysis to estimate errors in approximating the sums of series. These concepts are important in many areas of mathematics and science, including statistics, physics, and engineering.
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Find what X equals. 6 = -4(x - 2) + 2x
(show your work please!)
Answers
Answer:
1
Step-by-step explanation:
6 =-4(x-2)+2x
6 =-4x +8 +2x
6 - 8 =-4x +2x
-2 =-2x /-1
x=2:2
x=1
Let Vector f = ( 4, 3, 7 ) and vector g = (-1, 0, 2) . Which graph shows vector f + vector g
Answers
Represents the vector f + g. C.
The sum of two vectors, we simply add their corresponding components.
In this case,
f + g = (4 - 1, 3 + 0, 7 + 2) = (3, 3, 9)
So the resulting vector has components (3, 3, 9).
Now we need to find the graph that represents this vector.
Graphing a vector in three dimensions can be challenging, but we can use the following method:
Start at the origin (0, 0, 0) of the 3D coordinate system.
Move 3 units in the x-direction, 3 units in the y-direction, and 9 units in the z-direction.
Mark the endpoint of this displacement as the tip of the vector.
Option A has a similar direction, but it is longer than f + g.
Option B has the correct length, but it is pointing in the wrong direction.
Option D is pointing in the correct direction, but it is too short.
We only add the respective components of the two vectors to get their sum.
Thus, f + g = (4 - 1, 3 + 0, 7 + 2) = (3, 3, 9) in this situation.
The resultant vector has three (3), three (3), and nine (9).
We now need to identify the graph that this vector is represented by.
It might be difficult to graph a vector in three dimensions, however we can try the following approach:
Start at the 3D coordinate system's origin (0, 0, 0).
Move three units in the x, three units in the y, and nine units in the z directions.
Make a note of the vector's tip being the terminus of this displacement.
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