Let v, w∈Rn. If |‖v‖=‖w‖, show that v+w and v−w are orthogonal (perpendicular). Solution: Let's assume that |‖v‖=‖w‖. Then it implies that ‖v‖2=‖w‖2... (1)Now, let's consider (v+w).(v-w) =(v.v)+(v.-w)+(w.v)+(w.-w)(dot product formula). The cross terms will be zero as we consider vectors v and w to be orthogonal. So, (v.w)+(w.v). Now, we know that, v.w = |v||w|cosθw.v = |v||w|cosθ(w -ve angle will be taken as w.v will give negative value).
Therefore, v.w + w.v = |v||w|cosθ +|v||w|cosθ= 2|v||w|cosθ. From (1) above, we can say that, |v|=|w|. So, v.w + w.v = 2|v||w|cosθ = 2|v|2cosθ = 2‖v‖2cosθ=(v+w).(v-w) = 2‖v‖2cosθ. Here, we have two vectors (v+w) and (v-w), which makes an angle of θ with each other, from the above step it is evident that the dot product of these vectors is zero.
Hence, the given two vectors are orthogonal (perpendicular). Therefore, the given statement is proved.
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A box contains 12 cereal bars. The empty box weighs 1.75 oz. The box and cereal bars together weighs 18.55 oz. How much does each cereal bar weigh?
Answer:
Each cereal bar weighs 16.8 oz
Step-by-step explanation:
Multiply 12 times 1.4 to get 16.8
Add 16.8 and 1.75 to get 18.55
Hope this helps! Pls mark brainliest
OMG PLS HELP WITH THIS IM PANICKING OMG I GOT A F IN MATH MY MOM JUST YELLED AT ME IM CRYING PLS HELP WITH THS- PLSS TELL ME I ALREADY DID NUMBER 2 BTW SO NO NEED JUST PLS HELP IN THE BOTTOM :(
Answer:
3. The equivalent fractions are presented as follows;
[tex]\dfrac{2}{3} = \dfrac{2 \times 2}{3 \times 2} =\dfrac{4}{6}[/tex]
4. The equivalent fractions are presented as follows;
[tex]\dfrac{2}{3} = \dfrac{4 \times 2}{4 \times 3} =\dfrac{8}{12}[/tex]
5. The reason why it works is because [tex]\dfrac{1}{8} + \dfrac{1}{8} = \dfrac{1}{4}[/tex]
Step-by-step explanation:
3. Based on the shaded region of the given figures, the two fractions are equivalent
Mathematically, we can show the equivalency of the two fractions as follows;
[tex]\dfrac{2}{3} =\dfrac{2}{3} \times 1 = \dfrac{2}{3} \times \dfrac{2}{2} = \dfrac{2 \times 2}{3 \times 2} =\dfrac{4}{6}[/tex]
Therefore;
[tex]\dfrac{2}{3} = \dfrac{2 \times 2}{3 \times 2} =\dfrac{4}{6}[/tex]
4. From the shaded region of the given figure, the fractions are equivalent
Mathematically, we have;
[tex]\dfrac{2}{3} =\dfrac{2}{3} \times 1 = \dfrac{4}{4} \times \dfrac{2}{3} = \dfrac{4 \times 2}{4 \times 3} =\dfrac{8}{12}[/tex]
Therefore;
[tex]\dfrac{2}{3} = \dfrac{4 \times 2}{4 \times 3} =\dfrac{8}{12}[/tex]
5. The amount of salt required by the recipe = 1/4 teaspoon of salt
The measure of the spoon Jonas has = 1/8 teaspoon
When Jonas adds two 1/8 teaspoons, he gets;
[tex]\dfrac{1}{8} + \dfrac{1}{8} = \dfrac{1 + 1}{8} = \dfrac{2}{8} = \dfrac{2 \times 1}{2 \times 4} = \dfrac{2}{2} \times \dfrac{1}{4} = 1 \times \dfrac{1}{4} = \dfrac{1}{4}[/tex]
Therefore;
[tex]\dfrac{1}{8} + \dfrac{1}{8} = \dfrac{1}{4}[/tex]
Two 1/8 measuring teaspoons are equivalent to one 1/4 teaspoons and therefore Jonas is able to get the 1/4 teaspoons of salt the recipe asks for b combining two 1/8 measuring teaspoons of salt he has because 1/8 + 1/8 = 1/4.
a lamina occupies the part of the disk 2 2≤16 in the first quadrant and the density at each point is given by the function (,)=3(2 2).
A lamina occupies the region of a disk in the first quadrant where 2 ≤ r ≤ 16, and the density at each point is given by the function ρ(r, θ) = 3[tex](r^2).[/tex] Further analysis is required to determine the mass and other properties of the lamina.
The given information describes a lamina occupying a region in the first quadrant of a disk. The radial distance from the origin is limited to the range 2 ≤ r ≤ 16. The density of the lamina at any point within this region is determined by the function ρ(r, θ) = 3[tex](r^2)[/tex], where r represents the radial distance and θ represents the angle in the polar coordinate system.
To fully analyze the lamina, additional calculations are necessary. One important calculation is determining the mass of the lamina, which involves integrating the density function over the given region. By integrating the function ρ(r, θ) = 3[tex](r^2)[/tex] over the appropriate range of r and θ, we can find the total mass of the lamina. Additionally, other properties such as the center of mass or moment of inertia of the lamina could be determined by using appropriate formulas and integration techniques.
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Function 1 is represented by the equation y = -4/3x-2, and function 2 is represented by the
graph below.
FUNCTION 2
For which of the functions are all the output values less than -1?
A. Both functions
B. Only function 1
C. Only function 2
D. Neither functions
Part A is already answered
Part B asks: What is the length of the hypotenuse?
Answer:
52
Step-by-step explanation:
Using the Pythagorean theorem
a^2+b^2 = c^2
20^2 + 48^2 = c^2
400 +2304=c^2
2704=c^2
Taking the square root of each side
sqrt(2704) = sqrt(c^2)
52 = c
Answer:
[tex]hypotenuse^{2}[/tex] = [tex]altitude^{2}[/tex] + [tex]base^{2}[/tex]
[tex]hyp^{2}[/tex] = [tex]48^{2}[/tex] + [tex]20^{2}[/tex]
= 2304 + 400
= 2704
∴[tex]hyp[/tex] = [tex]\sqrt{2704}[/tex]
= 52
hope this answer helps you....
In a publication of a renowned magazine, it is stated that cars travel in
average at least 20,000 kilometers per year, but do you think the average actually
is minor. To test this claim, a sample of 100 car owners is asked
randomly selected to keep a record of the kilometers they travel. It would
If you agree with this statement, if the random sample indicates an average of 19,000
kilometers and a standard deviation of 3900 kilometers? Use a significance level of
0.05 and for its engineering conclusion use:
a) The classical method.
b) The P-value method as an auxiliary.
Both the classical method and the p-value method lead to the conclusion that the average distance cars travel per year is less than 20,000 kilometers,
a) t = -2.564
b) t = -2.564
How to thest the claim?To test the claim that the average distance cars travel per year is less than 20,000 kilometers, we can conduct a hypothesis test using the classical method and the p-value method.
a) The steps we need to follow are:
Step 1: Formulate the null and alternative hypotheses:
Null hypothesis (H₀): The average distance cars travel per year is 20,000 kilometers.
Alternative hypothesis (H₁): The average distance cars travel per year is less than 20,000 kilometers.
Step 2: Determine the test statistic:
Since we know the sample size (n = 100), the sample mean (x = 19,000 kilometers), and the sample standard deviation (s = 3900 kilometers), we can use the t-test statistic.
t = (x - μ₀) / (s / √n)
where μ₀ is the assumed population mean under the null hypothesis.
Step 3: Set the significance level:
The significance level is given as 0.05, which means we want to be 95% confident in our conclusion.
Step 4: Calculate the critical value:
Since the alternative hypothesis is one-tailed (less than), we need to find the critical t-value corresponding to a 0.05 significance level and degrees of freedom (df) = n - 1 = 99. From the t-distribution table or calculator, the critical t-value is approximately -1.660.
Step 5: Calculate the test statistic:
t = (19,000 - 20,000) / (3900 / √100)
t = -10 / (3900 / 10)
t ≈ -2.564
Step 6: Compare the test statistic with the critical value:
Since -2.564 is less than -1.660, the test statistic falls in the rejection region. We reject the null hypothesis.
Step 7: Make a conclusion:
Based on the classical method, since the test statistic falls in the rejection region, we conclude that the average distance cars travel per year is significantly less than 20,000 kilometers.
b) The P-value method:
Using the same test statistic t = -2.564 and the degrees of freedom (df) = 99, we can calculate the p-value. The p-value is the probability of observing a test statistic as extreme as the one obtained, assuming the null hypothesis is true.
From a t-distribution table or calculator, the p-value corresponding to t = -2.564 and df = 99 is approximately 0.0075 (or 0.75% if multiplied by 100).
Since the p-value (0.0075) is less than the significance level (0.05), we reject the null hypothesis. This suggests strong evidence that the average distance cars travel per year is significantly less than 20,000 kilometers.
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Help!! Please!!!!!!!
Answer:
E
Step-by-step explanation:
Solve the system of differential equations S x1 = – 5x1 + 0x2 – 16x1 + 322 X2' x1(0) = 1, X2(0) = 5 21(t) = = 22(t) - = X2
The solution to the system of differential equations is x₁(t) = e⁻⁵ˣ + 3e³ˣ and x₂(t) = 2e⁻⁵ˣ + 5e³ˣ
Let's solve the given system of differential equations: x₁' = -5x₁ + 0x₂ ...(1) x₂' = -16x₁ + 3x₂ ...(2)
To solve this system, we can rewrite it in matrix form. Let's define the vector X = [x₁, x₂] and the matrix A as:
A = [[-5, 0], [-16, 3]]
The system can then be written as X' = AX, where X' is the derivative of X with respect to time.
Now, let's find the eigenvalues and eigenvectors of matrix A. The eigenvalues are obtained by solving the characteristic equation det(A - λI) = 0, where I is the identity matrix.
A - λI = [[-5 - λ, 0], [-16, 3 - λ]]
det(A - λI) = (-5 - λ)(3 - λ) - 0(-16) = λ² + 2λ - 15 = (λ + 5)(λ - 3)
Setting the characteristic equation equal to zero, we find the eigenvalues: λ₁ = -5 λ₂ = 3
To find the corresponding eigenvectors, we substitute each eigenvalue back into the matrix A - λI and solve the system of equations (A - λI)v = 0, where v is the eigenvector.
For λ₁ = -5: A - (-5)I = [[0, 0], [-16, 8]]
Using Gaussian elimination, we can solve the system of equations to find the eigenvector corresponding to λ₁: -16v₁ + 8v₂ = 0 => -2v₁ + v₂ = 0 => v₁ = (1/2)v₂
Let v₂ = 2, then v₁ = 1. Therefore, the eigenvector corresponding to λ₁ is v₁ = [1, 2].
For λ₂ = 3: A - 3I = [[-8, 0], [-16, 0]]
Solving the system of equations, we find: -8v₁ = 0 => v₁ = 0
Thus, the eigenvector corresponding to λ₂ is v₂ = [0, 1].
Now, let's express the solution of the system in terms of the eigenvalues and eigenvectors.
X(t) = c₁e(λ₁t)v₁ + c₂e(λ₂t)v₂
Substituting the eigenvalues and eigenvectors we found earlier, we have: X(t) = c₁e⁻⁵ˣ[1, 2] + c₂e³ˣ[0, 1]
Using the initial conditions, x₁(0) = 1 and x₂(0) = 5, we can find the values of c₁ and c₂.
At t = 0: [1, 5] = c₁[1, 2] + c₂[0, 1] 1 = c₁ 5 = 2c₁ + c₂
Solving these equations, we find: c₁ = 1 c₂ = 3
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Complete Question:
Solve the system of differential equations
x₁' = – 5x₁ + 0x₂
x₂' = – 16x₁ + 3x₂
x₁(0) = 1, x₂(0) = 5
What is the solution to the system of
equations graphed below?
a.(-13,-20)
b.(-15,-22)
c.(-1,-8)
d.(-10,-15)
Answer:
(-13,-20)
Step-by-step explanation:
We know that the graph with x-intercept at (7,0) and y-intercept at (0,-7) is y = x-7
and the second equation is y = 2x+6
Therefore, we have two equations.
[tex] \large{ \begin{cases} y = x - 7 \\ y = 2x + 6 \end{cases}}[/tex]
Because both equations are equal (y=y)
x-7=2x+6
-7-6=2x-x
-13=x
We know the value of x, but not y-value. We substitute the value of x to get the value of y.
Substitute in any given equations which I will be substituting in the first equation.
y=-13-7
y = -20
Therefore when x = -13, y = -20
In coordinate form, we can write as (-13,-20).
If you have any questions, feel free to ask.
Andrea constructed a triangle. Angle 1 and 3 are the same size and angle 2 has a measurement of 70 degrees. What is the measurement of angle 1 and 3
Answer:
Step-by-step explanation:
By triangle sum theorem,
Sum of all angles of a triangle is 180°.
m∠1 + m∠2 + m∠3 = 180°
(m∠1 + m∠3) + m∠2 = 180°
2(m∠1) + 70° = 180° {Given → m∠1 = m∠3]
2(m∠1) = 110°
m∠1 = 55°
Therefore, m∠1 = m∠3 = 55°
round 3.060 to the nearest whole number.
Answer:
3
Step-by-step explanation:
dude kinda obv that its 3
Answer:
answer would be 3
Step-by-step explanation:
A UMass student is starting their junior year and has accumulated 60 credits so far. Their current cumulative average is a C, or a Grade Point Average (GPA) of 2.0. Their employer has a scholarship program for students who have GPAs of 2.3 or higher. This student wants to get that scholarship to help pay for their senior year. They plan on taking 15 credits each for the fall and spring semesters of their junior year.
a. Can they raise their cumulative average to 2.3 after completing 15 fall semester credits? What semester GPA would they need?
b. What average GPA would they need for their two junior year semesters combined (30 credits) to achieve their goal of a 2.3 cumulative GPA and 90 credits?
According to the information, we can infer that no, they cannot raise their cumulative average to 2.3 after completing 15 fall semester credits. On the other hand, they would need an average GPA of 3.25 for their two junior year semesters combined (30 credits) to achieve their goal of a 2.3 cumulative GPA and 90 credits.
How to calculate the new cumilative GPA?In order to calculate the new cumulative GPA, we need to consider both the current cumulative GPA and the GPA earned in the fall semester. Since the student's current cumulative GPA is 2.0 and they have already accumulated 60 credits, their total grade points earned so far would be 2.0 multiplied by 60, which equals 120 grade points.
To raise the cumulative GPA to 2.3, the student would need a total of 2.3 multiplied by (60 + 15) = 2.3 multiplied by 75 = 172.5 grade points by the end of the fall semester.
Since the student has already accumulated 120 grade points, they would need to earn an additional 52.5 grade points in the fall semester. To calculate the required semester GPA, we divide 52.5 by 15 credits, which gives us a required semester GPA of 3.5.
So, the student would need a semester GPA of 3.5 in order to raise their cumulative average to 2.3 after completing 15 fall semester credits.
What average gpa would they need for their two junior year semesters combined to achieve their goal?They would need an average GPA of 3.25 for their two junior year semesters combined (30 credits) to achieve their goal of a 2.3 cumulative GPA and 90 credits.
Explanation: To calculate the average GPA for the two junior year semesters, we need to consider the total grade points earned and the total number of credits taken.
Currently, the student has accumulated 60 credits and 120 grade points. In order to achieve a cumulative GPA of 2.3 after completing 90 credits, they would need a total of 2.3 multiplied by 90 = 207 grade points.
To calculate the required grade points for the two junior year semesters, we subtract the current grade points (120) from the desired total grade points (207), which gives us 207 - 120 = 87 grade points needed for the junior year.
Since the student plans to take 30 credits during their junior year, they would need to earn 87 grade points in those 30 credits. Dividing 87 by 30 gives us an average GPA of approximately 2.9 for the two junior year semesters.
According to the above, the student would need an average GPA of 3.25 (rounded up) for their two junior year semesters combined to achieve their goal of a 2.3 cumulative GPA and 90 credits.
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HURRY PLEASEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
Answer:
48 is the answer
Step-by-step explanation:
9*4 +. 6*2
36 + 12
48
math because im very bad at it
Can I get help with number 15 am stuck
In sarah classroom, There are 6 girls or every 2 boys. What is the ratio of boys to the total number of student
Answer:
3:1 (Girls:Boys)
6 divided by 2
HOW MANY TABLESPOONS ARE IN 400 MILLIMETERS? 1 TSP = 5mL
Answer:
80 tsp.
Step-by-step explanation:
400 divided by 5 is 80, so 80 tsp's.
Answer:
80 tsp.
Step-by-step explanation:
400 mL = 80 tsp
Write the integer that represents the opposite of each situation
100 feet above sea level
evaluate sum in closed form
f(x) = sin x + 1/3 sin 2x + 1/5 sin 3x + ....
The sum of given series is infinity ∑(n = 1) sin(nx)/(2n - 1).
What is sum of series?
A series' sum is the sum of the words or list of numbers that make up the series. If a series has a sum, it will be one integer (or fraction), such as 0, 1/2, or 99.
As given series is,
f(x) = sinx + sin(2x)/3 + sin(3x)/5+.....
The series function is trigonometric function such as sinx, sin(2x), sin(3x).
The nth term of series is sin(nx).
The coefficient of series is 1, 1/3, 1/5.
The coefficient of nth terms is 1/(2n - 1).
Sum of series is,
= infinity ∑(n = 1) sin(nx)/(2n - 1).
Hence, the sum of given series is infinity ∑(n = 1) sin(nx)/(2n - 1).
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How many sides do 4 pentagons and 3 nonagons have in all?
Answer:
47
Step-by-step explanation:
^
Which one would result an integer
Answer:
c is the only one that would result in an integer
Step-by-step explanation:
i hope this helps :)
Option c ∛ 27 would result in an integer.
what are integers?An integer is the number zero, a positive natural number or a negative integer with a minus sign. The negative numbers are the additive inverses of the corresponding positive numbers.
Given here, ∛27= 3 while the other options ∛60 is not an integer because 60 is not cubic number similarly for 9 , 18 are not cubic numbers and thus their subsequent cubic roots will not yield an integer.
Hence, Option c ∛ 27 would result in an integer.
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Plz answer quick need help. are the sums of double odd?
Which value of x satisfies the equation below? 1/2 (3x + 17) = 1/6 (8x-10)
Choice answers:
A. -61
B -55
C. -41
D-35
Find the total area.
Answer:
area = 164.52 cm²
Step-by-step explanation:
area = (12x6) + (12x3x0.5x2) + (3.14)(0.5)(6²) = 164.52 cm²
Which of these shapes is not a parallelogram? HELP ASAP 15 BRAINLY POINTS! TYSM GOD BLESS YOU AND YOUR FAMILY!
Answer:
The answer would be D____________________________________________________________
Why?
Trapezoids have only one pair of parallel sides; parallelograms have two pairs of parallel sides. A trapezoid can never be a parallelogram. The correct answer is that all trapezoids are quadrilaterals.
____________________________________________________________
What's a parallelogram?
Its a four-sided plane rectilinear figure with opposite sides parallel.
____________________________________________________________
Please don't be afraid to point out errors :)
____________________________________________________________
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¸ . ★ ° :. . • ° . * :. ☆
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¸ . ★ ° :. :. . ¸ . ● ¸ ° ¸. * ● ¸ °☆
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. * . . ° . ● ° .
° :. ° . ☆ . . • . ● .° °★ Not sure how to copy and paste? Just right click your mouse and choose copy in options, to release repeat the process and just paste it. No mouse? Select the text with your computer pad and use ctrl c to release, ctrl v. On mobile? Press on your screen and select the text, use the option copy and paste wherever you would like!
The correct shape which is not a parallelogram is shown in Option D.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
There are four shapes are shown.
Now, We know that;
In a parallelogram, there are two pair of parallel lines.
But In option D;
There are only one pair of parallel lines.
Hence, The correct shape which is not a parallelogram is shown in Option D.
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could any body help me with this
Answer:
1) 1.2 minutes
2) 8.3 laps
3) 25.2 minutes
solve the system of substitution y=-2x y=5x-21
Answer:
x = 3 is the answerStep-by-step explanation:
1. Write the equation.
y = -2x
y = 5x - 21
2. Substitute the values.
(-2x) = 5x - 21
3. Solve the equation.
-2x = 5x - 21
-5x - 5x
-7x = -21
4. Both negatives cancel
7x = 21
5. Divide both sides by 7
7x = 21
/7 /7
6. x = 3
7. Check the answer.
-2(3) = 5(3) - 21
-6 = 15 - 21
-6 = -6
x = 3 is the answerHope this helped,
Kavitha
P.S Sorry for taking so long.
Four minutes is what percent of an hour?
Answer
6 and 2/3 percent of an hour OR 6.666.... hour
I don’t know if you have to round or not but if it does just round
Step-by-step explanation:
Well 4 minutes of an hour is basically 4/60
4/60=1/15
1/15=x/100
solve the proportion by cross multiplying
100=15x
x=6.66666666
That is yoru percent
Help please I need this today
Answer:
a = 15
t = 5
Step-by-step explanation:
[tex] \frac{12}{a} = \frac{16}{20} \\ \\ 16a = 12 \times 20 \\ \\16 a = 240 \\ \\ a = \frac{240}{16} \\ \\ a = 15 \\ \\ \\ \frac{2}{8} = \frac{t}{20} \\ \\ 8t = 40 \\ \\ t = \frac{40}{8} \\ \\ t = 5[/tex]
Solve the following equation for x.
Answer:
If you just solve normally, you will get x=2 and x=-3. But if you plug these in to check your work, you will find that they are wrong. Your answer is no solution
Step-by-step explanation:
ln(2x+3)+ln(x-2)=ln(x^2-2x)
Rule: log(a) + log(b) = log(a*b)
ln( (2x+3)(x-2) ) = ln(x^2-2x)
Rule: If log(a) = log(b) then a = b
(2x+3)(x-2) = x^2 - 2x
2x^2-x-6=x^2-2x
x^2+x-6=0
Using Quadratic Formula:
x = 2 and x = -3
But, plugging these numbers back into the original equation is false!