show that if u and v are any vectors in r2, then u v2 ≤ (u v) 2 and hence u v≤u v. when does equality hold? give a geometric interpretation of the inequality.
Answers
The inequality \($uv^2 \leq (uv)^2$\) holds for any vectors u and v in \(R^2\). Equality holds when the vectors u and v are collinear or when one of them is the zero vector.
To prove the inequality \($uv^2 \leq (uv)^2$\), we can start by expressing the dot product of u and v in terms of their components. Let u = (\(u_1, u_2\)) and v = (\(v_1, v_2\)).
Then, the dot product of u and v is given by\(uv = u_1 v_1 + u_2 v_2\).
Now, consider the squared length of the projection of u onto v.
This can be calculated as \((uv)^2\) / \(||v||^2\), where ||v|| represents the length of vector v.
Since ||v||^2 = vv, we have \((uv)^2\) /\(||v||^2\) = \((uv)^2\) / (\($v \cdot v$\)).
On the other hand, the squared length of vector u can be expressed as \(u\cdot u = u1^2 + u2^2\).
Now, we can compare the two expressions: \(uv^2\) and \((uv)^2\) / (vv).
By substituting the expression for uv, we get \(uv^2\) = \((u_1 v_1 + u_2 v_2)^2\), and by substituting the expressions for \(||v||^2\) and uu, we get \((uv)^2\) / (vv) = (\(u_1^2 v_1^2 + u_2^2 v_2^2\)) / (\(v_1^2 + v_2^2\)).
It can be shown that \((u_1 v_1 + u_2 v_2)^2 \geq (u_1^2 v_1^2 + u_2^2 v_2^2)\) for any real numbers \(u_1, u_2, v_1, v_2\). Therefore, \(uv^2\) ≤ \((uv)^2\) / \((vv)\), which implies \(uv^2\) ≤ \((uv)^2\).
Equality holds in the inequality \($uv^2 \leq (uv)^2$\) when the vectors u and v are collinear or when one of them is the zero vector.
Geometrically, this inequality represents the fact that the squared length of the projection of vector u onto vector v is always less than or equal to the squared length of the projection of u onto v.
When the vectors are collinear, their projections coincide and the inequality becomes an equality. Similarly, when one of the vectors is the zero vector, its projection onto any other vector is zero, resulting in equality as well.
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Related Questions
A square sits with its lower left corner at the orgin the square is translated five units up and seven units to the right what are the coordinates of the lower left corner of the square after translation
Answers
Answer:
\((2,0)\)
Step-by-step explanation:
Given
The initial lower left corner of the square at:
\((x,y) = (0,0)\)
Required
Its new position after 5 units left and 7 units right translations
When a point is translated to the left by a units, the rule is:
\((x,y) ==> (x-a,y)\)
In this case:
\(a = 5\)
So:
\((0,0) ==> (0-5,0)\)
\((0,0) ==> (-5,0)\)
So, the new point is: (-5,0)
When a point is translated to the right by b units, the rule is:
\((x,y) ==> (x+b,y)\)
In this case:
\(b = 7\)
So:
\((-5,0)= (-5+7,0)\)
\((-5,0)= (2,0)\)
So, the new point is: (2,0)
What is the image of the point (5, -3) after a rotation of 90°
counterclockwise
about the origin?
Answers
So, the new x-coordinate will be -(-3) = 3, and the new y-coordinate will be 5.
Therefore, the image of the point (5, -3) after a rotation of 90° counterclockwise about the origin is (3, 5).
Pls help me out. :)..
Answers
Answer:
1402 tiles
Step-by-step explanation:
so in order to find the amount of glass tiles we will first need to find the surface area of the box
so the surface area formula for a rectangular prism is
2 (wl * hw* hl)
w- width
h- height
l- length
so the width would be 18centimeters
the height would be 7 cm
and the length would be 23 cm
therefore if we plug in the measurements into the equation it will give us a total surface area of
1402cm^2
therefore if Dimitri wants to cover the box with 1cm glass tiles then we take the 1402cm and divide it by 1cm leaving us with the same answer of
1402 tiles
Can someone help me with this btw the bottom one is 5 1/2
Answers
V= blh
So all u have to do is multiply all of the side values!
V=5.5(3)(6.33)
V≈104.49945
Rounded is 104.5 :))
Select all of the following that are ordered pairs of the given function.
f(x) = 3 - 2x
(-2,-1)
(-1,5)
(0, 3)
(1, 0)
(2,-1)
Answers
Answer:
(-1, 5), (2, -1), (0, 3)
$4.50 for 10 ounces of trail mix
Answers
Answer:
45¢ for 1 ounce of trail mix
is that what you're looking for?
Step-by-step explanation:
4.50÷10=.45
What is the sum of the 10th square number and the 2nd cube number?
Answers
Answer:
100 + 27
127
bcoz 10th square no root is 10 and 2nd cube no root is 3
Citrix Apps Apps CANVAS > Home EPB Intranet 7. -/2 points RogaCalcET4 13.5.017.Tutorial. Find r(t) and v(t) given a(t) and the initial velocity and position. a(t) = tk, v(0) = 4i, r(0) = 2; v(t) = r(t) = Additional Materials Tutorial +-12 points RogaCalcETA 19 rann
Answers
The position value, r(t) is equals to the (t³/6)k + 2j and velocity value, v(t) is equals to ( t²/2 )k + 4i , for a(t) = tk, v(0) = 4i, r(0) = 2j.
Acceleration is defined as the rate of change of the velocity of an object with respect to time. Accelerations are vector quanty.
a = dv/dt
We have the following informations are available,
Initial velocity, v(0) = 4i
Initial position, r(0) = 2j
Acceleration at any time "t",
a(t) = tk
we have to determine the value of v(t) and r(t).
As we know, a(t) = dv(t)/dt = tk
integrating the above equation ,
v(t) = ∫tk dt = ( t²/2 )k + c
at t = 0 , v(0) = 0 + c = 4i ( since, v(0) = 4i
=> c = 4i
So, v(t) = ( t²/2 )k + 4i
Also, velocity is calculated by derivative of postion (r) with respect to time.
=> v(t) = dr(t) /dt
=> r(t) = ∫ v(t) dt
=> r(t) = ∫ ( t²/2 )k dt
integrating value of the right hand side,
r(t) = ( t³/2×3 )k +d
= (t³/6)k + d
At t = 0, r(0) = (0/6)k + d
=> r(0) = d = 2j
so, r(t) = (t³/6)k + 2j
Hence, the required position and velocity are
(t³/6)k + 2j and ( t²/2 )k + 4i.
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I need help on a problem
Answers
b. Cannot be determined
Explanation:b. The triangle have one congruent angles and one congruent sides, this is not enough information to conclude that they are congruent triangles.
Pls help this is confusing :/
Use matrices to determine the coordinates of the vertices of the rotated figure. Then graph the pre-image and the image of the same coordinate grid. \(Rot_{270}\) for ΔUVW with vertices U(-3,8), V(8,4), and W(6,-8) This is for pre-calculus!
Answers
The coordinates of triangle U'V'W' include U'(8, 3), V'(4, -8) and W'(-8, -6) and this is represented by graph A shown in the image attached below.
What is a transformation?A transformation can be defined as the movement of a point on a cartesian coordinate from its original (initial) position to a new location.
The types of transformation.In Geometry, there are different types of transformation and these include the following:
DilationReflectionRotationTranslationBased on the information provided, triangle UVW would be rotated counterclockwise through an angle of 270 degree at origin to produce triangle U'V'W', we have:
\(\left[\begin{array}{ccc}0&1\\-1&0\end{array}\right]\)
Therefore, the image of triangle UVW would be given by this matrix:
\(\left[\begin{array}{ccc}0&1\\-1&0\end{array}\right]\) \(\left[\begin{array}{ccc}-3&8&6\\8&4&-8\end{array}\right]\)
Image = \(\left[\begin{array}{ccc}8&4&-8\\3&-8&-6\end{array}\right]\)
Based on the image above, we can logically deduce that the coordinates of triangle U'V'W' include U'(8, 3), V'(4, -8) and W'(-8, -6) and this is represented by graph A shown in the image attached below.
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Answer:
hi friend
Step-by-step explanation:
Solve the equation for x.
Answers
answer:
x=-2
Step-by-step explanation:
my suggestion for this problem and all problems like it would to turn everything into 6ths. 1/3 would become 2/6. and 2 would become 12/6. leave the negatives. now you'll add 2/6 to -12/6. which will equal 10/6. divide 10/6 by 5/6 and youll get -2. x=-2
Answer:
x = -2
Step-by-step explanation:
● (5/6)x - (1/3) = -2
Add (1/3) to both sides
● (5/6)x - (1/3) + (1/3) = -2 + (1/3)
-2 is (-6/3)
● (5/6)x = (-6/3) + (1/3)
● (5/6)x = -5/3
Move 5/6 to the other side and switch the places of 5 and 6
● x = (-5/3) × (6/5)
● x = -30/15
● x = -2
DONT ANSWER IF YOU DONT KNOW AND NO LINKS PLEASE
Answers
Answer:
a reflection across the y axis.
Step-by-step explanation:
If you reflect (flip) the equation over the y axis it will stay the same.
pleaseeee asapppppp asapppppp
Answers
Answer:
7.07
Step-by-step explanation:
Angle M is 63 degrees because all 3 angles must add to 180.
Now use the law of sines:
sin(90)/x = sin(63)/6.3
Cross multiply: 6.3 sin (90) = x sin (63)
Divide by sides by sin (63): (6.3 sin (90))/sin (63) = x
Use a calculator: 7.07
Beyonce is solving a system of equations: 2x – 3y = -2
4x + y = 24
She decides to use the elimination method to find the solution. She multiplies the second equation by 3 and adds it to the first equation to find 14x = 70, showing her that x = 5. Beyonce finds that y = 4.
Step 3: 4x + y = 24
4(5) + y = 24
20 + y = 24
-20 -20
y = 4
Step 1: 3(4x + y = 24) → 12x + 3y = 72
Step 2: 2x – 3y = -2
Solution: (5, 4)
+ 12x + 3y = 72
14x = 70
→ x = 5
Thinking about this process, Beyonce says out loud, “There are lots of ways I could go about solving this problem. I could do the method above or I could multiply the first equation by -2 and add the second.
Step 3: 4x + y = 24
4x + 4 = 24
-4 -4
4x = 20
x = 5
Step 1: -2(2x – 3y = -2) → -4x + 6y = 4
Solution: (5, 4)
Step 2: -4x + 6y = 4
+ 4x + y = 24
7y= 28
→ y = 4
“I seem to find that there is only one solution to the two equations, but I wonder if I will get the same solution if I use a different method?”
Explain how you know that a system of equations has no solution.
Answers
Answer:
Step-by-step explanation:
A system of equations has no solutions if you end up with a contradiction. For example, the system of two equations
y = 2x + 4
y = 2x
If you subtract the second equation from the first, you get
0 = 4
an obvious contradiction. If you were to graph the equations, you would see that the lines are parallel. They never intersect, so there is no point that satisfies both equations.
A flower bed measuring 3m by 1.5m is surrounded by a path 1m wide. Find the area of the path.
Answers
3.4 Step-by-step explanation:
The area of path is 13.5 m².
What is area of rectangle?The area can be defined as the amount of space covered by a flat surface of a particular shape. It is measured in terms of the "number of" square units (square centimeters, square inches, square feet, etc.) The area of a rectangle is the number of unit squares that can fit into a rectangle.
Thus, the formula for the area, 'A' of a rectangle whose length and width are 'l' and 'w' respectively is the product "l × w".
length of flower bed = 3m
width of flower of bed = 1.5 m
length of bed along path = 3 + 1+ 1 = 5m
width of flower bed = 1.5 + 1+ 1 = 3.5 m
So, area of flower bed along path = 5 x 3.5 = 17.5 m²
and, area of flower bed = 3 x 1.5 = 4.5 m²
Thus, area of path = 17.5 - 4.5 = 13.5 m²
hence, the area of path is 13.5 m².
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if the volume of a ball is 32,490 cubic millimeters, what is the volume of the ball in cubic centimeters?
Answers
The volume of ball is 32.49 cubic centimeter.
What is volume ?
Every three-dimensional item takes up space in some way. The volume of this area is what is used to describe it. The area occupied within an object's three-dimensional bounds is referred to as its volume. The object's capacity is another name for it.
Here ,
The volume of a ball = 32490 cubic millimeters
To convert into cubic millimeter into cubic centimeter by dividing by 1000, Then,
=> 32490/1000
=> 32.49 cubic centimeter.
Hence the volume of ball in cubic centimeter is 32.49 .
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Find the volume of each rectangular prism from the given parameters.
length = 19 in ; width = 17 in ; height = 13 in
best answer gets 55 points
Answers
19 x 17 x 13 = 4183 in³
The volume of the rectangular prism is 4183 cubic inches.
a random sample of 150 u.s. adults is composed of 100 people from non-rural areas and 50 people from rural areas. estimate the confidence interval for the fraction of the population that is non-rural with 99% confidence. what is the lower limit of the confidence interval?
Answers
With 99% confidence, we can estimate that the fraction of the population that is non-rural is between 0.60 and 0.74.
To estimate the confidence interval for the fraction of the population that is non-rural with 99% confidence, we need to use the following formula:
Point estimate +/- (critical value * standard error)
Where the point estimate is the sample proportion of non-rural adults (100/150 = 0.67), the critical value is the z-score corresponding to a confidence level of 99% (2.575), and the standard error is the standard deviation of the sampling distribution of the sample proportion (which is equal to the square root of [(sample proportion * (1 - sample proportion)) / sample size]).
Plugging in the values, we get:
0.67 +/- (2.575 * sqrt[(0.67 * (1 - 0.67)) / 150])
Solving this equation gives us a lower limit of 0.60 for the confidence interval.
Thus, with 99% confidence, we can estimate that the fraction of the population that is non-rural is between 0.60 and 0.74.
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The first term of a geometric sequence is 1 and its common ratio is 6. What is the sixth term?
a. 31
b. 3176
c. 7776
d. 46,656
Answers
Answer:
c. 7776
Step-by-step explanation:
1, 6, 36, 216, 1296, 7776
Therefore, c. 7776
I hope this helped and have a good rest of your day!
Work out the size of angle X.
Answers
each face of a cube is assigned a different integer. then each vertex is assigned the sum of the integer values on the faces that meet at the vertex. finally, the vertex numbers are added. what is the largest number that must divide the final sum for every possible numbering of the faces?
Answers
The largest number is 4 which divides the final sum of every possible numbering of the faces.
As per the given problem,
Each face of a cube is assigned a different integer. Let the assigned integer be as follows:
face 1 - aface 2 - bface 3 - cface 4 - dface 5 - eface 6 - fThen each vertex is assigned the sum of the integer values on the faces that meet at the vertex. Later on, the vertex numbers are added as below:
vertex 1 ⇒ faces = a, d, e and sum = a + d + evertex 2 ⇒ faces = a, b, e and sum = a + b + evertex 3 ⇒ faces = b, c, e and sum = b + c + evertex 4 ⇒ faces = c, d, e and sum = c + d + evertex 5 ⇒ faces = a, d, f and sum = a + d + fvertex 6 ⇒ faces = a, b, f and sum = a + b + fvertex 7 ⇒ faces = b, c, f and sum = b + c + fvertex 8 ⇒ faces = c, d, f and sum = c + d + fTotal sum = a + d + e + a + b + e + b + c + e + c + d + e + a + d + f + a + b +
f + b + c + f + c + d + f
= 4a + 4b + 4c + 4d + 4e + 4f
sum = 4( a + b + c + d + e + f)
Therefore, 4 is the largest number that must divide the final sum for every possible numbering of the faces.
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Identify the surface with the given vector equation. r(s, t) = (s cos(t), s sin(t), s) circular paraboloid O elliptic cone O hyperbolic paraboloid O plane O circular cone X
Answers
The surface with the given vector equation, r(s, t) = (s cos(t), s sin(t), s), is a circular cone.
The vector equation r(s, t) = (s cos(t), s sin(t), s) represents a surface in three-dimensional space. Let's analyze the equation to determine the nature of the surface.
In the equation, we have three components: s, cos(t), and sin(t). The presence of s indicates that the surface expands or contracts radially from a central point. The trigonometric functions cos(t) and sin(t) determine the angle at which the surface extends in the x and y directions.
By observing the equation closely, we can see that as s increases, the radius of the surface expands uniformly in all directions, while the height remains constant. This behavior is characteristic of a circular cone. The circular base of the cone is defined by s cos(t) and s sin(t), and the vertical component is determined by s.
Therefore, the surface described by the vector equation r(s, t) = (s cos(t), s sin(t), s) is a circular cone.
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What is the equation of the trend line in the scatter plot?
Use the two yellow points to write the equation in slope-intercept form. Write any coefficients as integers, proper fractions, or improper fractions in simplest form.
Answers
Answer:
y = 7/5x - 21/5
Step-by-step explanation:
points to use to find slope: (3,0) and (8,7)
slope = 7-0 / 8-3
slope = 7/5
find y-intercept (or 'b')
0 = 3(7/5) + b
0 = 21/5 + b
-21/5 = b
b = -4 1/5
11. A work day is 8 hours. You earn $48 for one work day What is the hourly pay rate? b. At that rate how much would you earn in 5 hours? Ker notes
Answers
Answer:
You divide 48 and 8 to get 6 meaning you get $6 an hour.
6 x 5 = $30, soin 5 hours you get $30
Step-by-step explanation:
Answer:
A. You would earn 6 dollars an hour.
B. You would earn 30 dollars working for 5 hours.
what is the raw score that corresponds to a z-score of -2.25 in a population with a mean of 30 and a standard deviation of 6?
Answers
The raw score corresponding to a z-score of -2.25 in a population with a mean of 30 and a standard deviation of 6 is 16.5.
The z-score is the number of standard deviations from the mean, so it is a standardized score.
To solve for a raw score corresponding to a given z-score, we can use the formula:
z = (x - μ) / σ
where z is the z-score,
x is the raw score,
μ is the mean,
and σ is the standard deviation.
Rearranging the formula, we can solve for x : x = μ + (z * σ)
Substituting the given values: μ = 30, σ = 6,
and z = -2.25x
= 30 + (-2.25 * 6) x
= 30 - 13.5 x
= 16.5
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(3+6n³ + 3n) - (3n+3n³-2n¹)
Answers
Answer:
3n³ + 2n + 3
Step-by-step explanation:
(3+6n³ + 3n) - (3n+3n³-2n¹)
3 + 6n³ + 3n - 3n - 3n³ + 2n
3n³ + 2n + 3
Evaluate. Write your answer as a whole number or as a simplified fraction.
Submit
Answers
Answer: 1/121
Step-by-step explanation:
Any number raised to the power of 0 is 1
11^2 = 121
a certain species of deer Is to be introduced Into a forest Wildlife experts estimate The population will grow to P(t)= (988)3 1/2 where t where represents The number of years from the time of introduction.How long will it take for the population to reach 26676 deer according to this model?
Answers
Using exponential function it will take approximately 2.6 years for the population to reach 26676
Exponential FunctionExponential function, as its name suggests, involves exponents. But note that, an exponential function has a constant as its base and a variable as its exponent but not the other way round (if a function has a variable as the base and a constant as the exponent then it is a power function but not an exponential function). An exponential function can be in one of the following forms.
The formula given in this question is
P(t) = 988(3.5)ˣ
x = number of years from time of introductionThe time it will take to reach a population of 26676 is
26676 = 988(3.5)ˣ
x = 2.6 years
It will take approximately 2.6 years
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2(3a + 2) = – 8
show your work
Answers
Answer:
a=-2
Step-by-step explanation:
\(2(3a + 4) = - 8 \\( 2 \times 3a = 6a)(2 \times 2 = 4) \\ - 8 - 4 = - 12 \\ 6a \div - 12 = - 2\)
is it true?
32.55 rounds to 32
Answers
Answer:
False. 32.55 rounds to 33
Step-by-step explanation:
If the number after the decimal is larger than 4, then you round up.
Answer:
This isn't true.
Step-by-step explanation:
This isn't true because the fives (5) in 32.55 causes the number to be rounded up to 32. If this doesn't make sense, please message me and I can help you further!