Solution :
Given :
a = (1, 2, 3, 4) , b = ( 4, 3, 2, 1), c = (1, 1, 1, 1) ∈ [tex]R^4[/tex]
a). (a.2c)b + ||-3c||a
Now,
(a.2c) = (1, 2, 3, 4). 2 (1, 1, 1, 1)
= (2 + 4 + 6 + 6)
= 20
-3c = -3 (1, 1, 1, 1)
= (-3, -3, -3, -3)
||-3c|| = [tex]$\sqrt{(-3)^2 + (-3)^2 + (-3)^2 + (-3)^2 }$[/tex]
[tex]$=\sqrt{9+9+9+9}$[/tex]
[tex]$=\sqrt{36}$[/tex]
= 6
Therefore,
(a.2c)b + ||-3c||a = (20)(4, 3, 2, 1) + 6(1, 2, 3, 4)
= (80, 60, 40, 20) + (6, 12, 18, 24)
= (86, 72, 58, 44)
b). two vectors [tex]\vec A[/tex] and [tex]\vec B[/tex] are parallel to each other if they are scalar multiple of each other.
i.e., [tex]\vec A=r \vec B[/tex] for the same scalar r.
Given [tex]\vec p[/tex] is parallel to [tex]\vec a[/tex], for the same scalar r, we have
[tex]$\vec p = r (1,2,3,4)$[/tex]
[tex]$\vec p = (r,2r,3r,4r)$[/tex] ......(1)
Let [tex]\vec q = (q_1,q_2,q_3,q_4)[/tex] ......(2)
Now given [tex]\vec p[/tex] and [tex]\vec q[/tex] are perpendicular vectors, that is dot product of [tex]\vec p[/tex] and [tex]\vec q[/tex] is zero.
[tex]$q_1r + 2q_2r + 3q_3r + 4q_4r = 0$[/tex]
[tex]$q_1 + 2q_2 + 3q_3 + 4q_4 = 0$[/tex] .......(3)
Also given the sum of [tex]\vec p[/tex] and [tex]\vec q[/tex] is equal to [tex]\vec b[/tex]. So
[tex]\vec p + \vec q = \vec b[/tex]
[tex]$(r,2r,3r,4r) + (q_1+q_2+q_3+q_4)=(4, 3,2,1)$[/tex]
∴ [tex]$q_1 = 4-r , \ q_2=3-2r, \ q_3 = 2-3r, \ q_4=1-4r$[/tex] ....(4)
Putting the values of [tex]q_1,q_2,q_3,q_4[/tex] in (3),we get
[tex]r=\frac{2}{3}[/tex]
So putting this value of r in (4), we get
[tex]$\vec p =\left( \frac{2}{3}, \frac{4}{3}, 2, \frac{8}{3} \right)$[/tex]
[tex]$\vec q =\left( \frac{10}{3}, \frac{5}{3}, 0, \frac{-5}{3} \right)$[/tex]
These two vectors are perpendicular and satisfies the given condition.
c). Given terminal point is [tex]\vec a[/tex] is (-1, 1, 2, -2)
We know that,
Position vector = terminal point - initial point
Initial point = terminal point - position point
= (-1, 1, 2, -2) - (1, 2, 3, 4)
= (-2, -1, -1, -6)
d). [tex]\vec b = (4,3,2,1)[/tex]
Let us say a vector [tex]\vec d = (d_1, d_2,d_3,d_4)[/tex] is perpendicular to [tex]\vec b.[/tex]
Then, [tex]\vec b.\vec d = 0[/tex]
[tex]$4d_1+3d_2+2d_3+d_4=0$[/tex]
[tex]$d_4=-4d_1-3d_2-2d_3$[/tex]
There are infinitely many vectors which satisfies this condition.
Let us choose arbitrary [tex]$d_1=1, d_2=1, d_3=2$[/tex]
Therefore, [tex]$d_4=-4(-1)-3(1)-2(2)$[/tex]
= -3
The vector is (-1, 1, 2, -3) perpendicular to given [tex]\vec b.[/tex]
find the midpoint between this pair of points (-4,10) and (14,0)
Explanation:
The x coordinates are -4 and 14. They add up to -4+14 = 10. Then this cuts in half to get 10/2 = 5. This is the x coordinate of the midpoint.
We'll follow this same idea for the y coordinates as well.
10+0 = 10 which cuts in half to 10/2 = 5, and this is the y coordinate of the midpoint.
Therefore, the midpoint is (5,5).
Coincidentally, the x and y coordinates are the same for the midpoint (both are 5). This won't always happen.
Find the value of x. Round the length to the nearest tenth.
Answer:
5.1
Step-by-step explanation:
[tex]tan \ 25 = \frac{opposite \ side}{adjacen \ side}\\\\tan \ 25 =\frac{x}{11}\\\\0.4663 =\frac{x}{11}[/tex]
0.4663 * 11 = x
x = 5.1293
x = 5.1
a triangle with 2 equal sides is called
Answer:
isosceles triangle.
Step-by-step explanation:
Answer:
Isosceles triangle
Step-by-step explanation:
In isosceles triangle, two sides are equal and the angles opposite to these equal angles are also equal. The third side of the triangle is called base
Choose the scatter plot that shows a negative correlation.
Click on the graph until the correct graph appears.
One way to test for one to one functions is to...
Group of answer choices :
restrict the function's domain.
put the equation into the quadratic formula.
perform a horizontal line test.
perform a vertical line test.
Answer:
Perform a horizontal line test
please help me and show work on a piece of paper
Let's start with proving that side AB is parallel to side CD
First off, let's find the slope of line AB. So we'll use the slope formula.
[tex]A = (x_1,y_1) = (2,-1)\\\\B = (x_2,y_2) = (1,3)\\\\m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\\\\m = \frac{3 - (-1)}{1 - 2}\\\\m = \frac{3 + 1}{1 - 2}\\\\m = \frac{4}{-1}\\\\m = -4\\\\[/tex]
The slope of AB is -4.
Now let's find the slope of line CD.
[tex]C = (x_1,y_1) = (6,5)\\\\D = (x_2,y_2) = (7,1)\\\\m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\\\\m = \frac{1 - 5}{7 - 6}\\\\m = \frac{-4}{1}\\\\m = -4\\\\[/tex]
The slope of CD is also -4.
Recall that parallel lines always have equal slope (but different y intercepts).
This proves that AB is parallel to CD.
--------------------------
We'll follow the same steps to show that AD is parallel to BC.
Find the slope of AD
[tex]A = (x_1,y_1) = (2,-1)\\\\D = (x_2,y_2) = (7,1)\\\\m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\\\\m = \frac{1 - (-1)}{7 - 2}\\\\m = \frac{1 + 1}{7 - 2}\\\\m = \frac{2}{5}\\\\[/tex]
Now find the slope of BC
[tex]B = (x_1,y_1) = (1,3)\\\\C = (x_2,y_2) = (6,5)\\\\m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\\\\m = \frac{5 - 3}{6 - 1}\\\\m = \frac{2}{5}\\\\[/tex]
We get 2/5 each time, so these lines are parallel as well.
--------------------------
In short, the first section shows that AB is parallel to CD. The second section proved that AD is parallel to BC.
Because we have two pairs of parallel opposite sides, we can say that ABCD is a parallelogram.
Edit: I have attached the diagram below.
please help!! 10 points! Need 2 and 5 answered!
Answer:
2. given
5. substitution
Step-by-step explanation:
that the perimeter is 12 is simply a given fact. in the same way as the shape of the object is defined as square.
nothing was calculated, this information is simply there at the beginning, is "given" to us.
step 5 simply takes step 3 and substitutes all sides by the same side, as (per definition) all sides in a square are equal. so, every side can stand in for every other side.
Estimate the answer to
(5.9 x 10^4) x (9.8 x 10^5)
Give your answer in standard form.
Answer:
5•782×10^-5
Step-by-step explanation:
(5•9×10^4((9•8×10^5)
(59×10^-1×10^4)×(98×10^-1×10^5)
(59×98) (10^-1×10×4×10-1×10^5)
5782×10^-1+^4+^-1+^5
578×10^7
5•782×10^7-^2
5•782×10^-5
Thanks Hope It Help
A) 78.5 cm
B) 4.7cm
C) 5.8cm
D) 23.0 cm
Answer:
The answer to this would be B.) 4.7 cm
Step-by-step explanation:
Arch = (Angle/360)(2[tex]\pi[/tex]r)
= (45/360)(2[tex]\pi[/tex]6)
= (0.125)(12[tex]\pi[/tex])
=1.5[tex]\pi[/tex]
≈4.71cm (assuming [tex]\pi[/tex] = 3.14)
Since an arch of a circle is just part of the circumference, you just need to multiply the circumference by the measure of the arch's angle divided by 360 (the angle measure of a circle).
A model is an estimate or approximation of a data set. In the real world, there are too many variables to know exactly what will happen. However, if a model is a good fit, then it can be used to make predictions about what will happen. With a useful model, a predicted value should be a good estimate of an observed value. What did you include in your response? Check all that apply. A model is an estimate of a data set. A useful model can be used to make predictions. Predictions should be a good estimate of what will actually happen.
Answer:
A model is an estimate of a data set.
A useful model can be used to make predictions.
Step-by-step explanation:
There are various models used in mathematics which are used to predict variables. The regression model is used to predict the relation between two variables. There are various other models which help to predict the future events and outcome of these events.
20
f(x)
g(x)
The tables given are for the linear functions f(x) and
g(x). What is the input value for which f(x) = g(x) is true?
x -2
-2
X=
3
4
-1
3
-1
3
0
3
0
2.
1
3
1
1
N
3
N
0
3
3
3
-1
Answer:
1 is the wrong answer i dont know the answer plz help him
Which parabola(s) have an axis of symmetry of x = -1
Answer:
b,c
Step-by-step explanation:
The axis of symmetry is along the vertex
a has a vertex at x=1
b has a vertex at x = -1
c has a vertex at x=-1
d has a vertex at x =2
Explanation:
Notice how the vertical line through the center of these parabolas is at x = -1. In other words, the vertical line cuts through -1 on the x axis. Reflecting one half of the parabola over this line will result in the other half.
similar right triangles, i need help with this please
Answer:
Step-by-step explanation:
the answer is a)
Use the graph below for this question:
what is the average rate of change from x = 0 to x = 2?
a) 3
b) -1
c) 2
d) -2
The average rate of change is the same as the slope. Even though this graph is a parabola, we still use the slope formula with the corresponding points. Had the question asked for the instantaneous rate of change, we would have needed to find a derivative.
Slope formula: (y2 - y1) / (x2 - x1)
Point 1: (0,4)
Point 2: (2,8)
(8 - 4) / (2 - 0)
4 / 2
2
The average rate of change from x = 0 to x = 2 is 2.
Hope this helps!
can i get some help please
Answer:
(1,-5)
Step-by-step explanation:
4x + 2y = -6
-3x -2y = 7
in this case you can just add the two items together
x = 1
4(1) + 2y = -6
2 y = -10
y = -5
(2/3)^x-1=27/8, Find x. Please, a step-by-step explanation.
Answer:
x=-2
Step-by-step explanation:
(2/3)^x-1=27/8
Rewriting the right hand side
(2/3)^x-1=(3/2)^3
We know that a^-b = 1/a^b Hint: { 1/(3/2) = 2/3}
(2/3)^x-1=(2/3) ^-3
The bases are the same, so the exponents are the same
x-1 = -3
x -1+1 = -3+1
x = -2
Point P is the incenter of DEF. Point P is the point of concurrency of the angle bisector. Find AP.
Answer:
5
Step-by-step explanation:
because line PF DIVIDED LINE BPA INTO TWO EQUAL HALF THEREFORE LINE BP IS EQUAL TO LINE AB WHICH IS EQUAL TO 5
The required value of the PV is 12 in the triangle DEF where p is the incenter.
Given; Point P is the incenter of ΔDEF. Point P is the point of concurrency of the angle bisector. PV to be determined.
What is incenter?The incenter of a triangle is the intersection point of the internal angle bisectors of the triangle.
In expressions, it can be clarified as the point where the internal angle bisectors of the triangle and perpendicular distance from this point to any of the sides of triangle is the same.
Since in the triangle FGH, P is the incenter and PW, PV,PX is perpendicular distance to the sides HF, GF and GH respectively.
From image given,
PW = 12
So by the property of the incenter, perpendicular distance from this point to any of the sides of triangle is the same.
PW = PV
PV = 12
Therefore, the required value of the PV is 12 in triangle DEF where p is the incenter.
Learn more about incenter here:
brainly.com/question/27868922
#SPJ2
Tanya has a garden with a trench around it. The garden is a rectangle with a length of 2 1/2 m and width 2 m The trench and garden together make a rectangle with length 3 1/2 and width 3m
Find the area of just the trench.
Answer:
Area of the rectangular trench = 1 square meter
Step-by-step explanation:
length of the garden = 2 1/2 m width of the garden = 2 m
length of both garden and trench = 3 1/2 m
width of both garden and trench = 3m
Length of the trench = Difference in length
= 3 1/2 m - 2 1/2 m
= 7/2 - 5/2
= (7 - 5) / 2
= 2/2
= 1 m
Length of the trench = 1 m
Width of the trench = difference in width
= 3 m - 2 m
= 1 m
Width of the trench = 1 m
Area of the rectangular trench = length × width
= 1 m × 1 m
= 1 m²
Or
1 square meter
Area of the rectangular trench = 1 square meter
SMART PPL PLEASE LOOK AT THIS I BEG YOU AND I WILL DO ANYTHING U WANT!!
Answer:
the real answer is
2X/19
Solve for x.
x - 42 = 98 – 9x
x = [?]
Answer:
x = 14
Step-by-step explanation:
x - 42 = 98 – 9x
Add 9x to each side
x - 42+9x = 98 – 9x+9x
10x-42 = 98
Add 42 to each sid
10x-42+42 = 98+42
10x = 140
Divide by 10
10x/10 =140/10
x = 14
8y-(-9.3z+9) how do i do this
What is the slope of the line that contains these points? x- 31, 36,41,46 y- 10,8,6,4
[tex] - \frac{2}{5} [/tex]
The result of dividing (x3 + x2 – 22x – 40) by
(x – 5) is x2 + 6x + 8. Select all of the factor(s) of x2 + 6x + 8.
(x – 4)
(x + 4)
(x – 2)
(x + 2)
Answer: b and d
Step-by-step explanation:
edge21/22
At the end of a game, Fran had 14 points and Calvin had -22 points. How many points did Fran win? A: -36 B: 8 C: -8 D: 36
Answer:
36
Step-by-step explanation:
Number of points Fran won = points Fran had - points Calvin had
14 - (-22)
= 14 + 22
= 36
justificantion for x=8/15
as according to law if a number is multiplied at one side then it will be divide to other side
so
x=8/15Step-by-step explanation:
plz mark me asbrain liest
Find the least number when divided by 75,125 and 225 Leaves a remainder 6 in each case.
Answer:
676,131
Step-by-step explanation:
L.C.M of 75,125 and 225,
=>676,125
Leaves a remainder 6 means,
=>676,125 + 6
=>676,131
That means,
When you divide 676,131 by 75,125 or 225,
You get remainder 6 .
That is why we add 6 to the L.C.M
You buy tomato sauce for $0.37 per ounce. One portion of potato gnocchi requires 3.5 ounces of tomato sauce. How much does the sauce for one portion cost? Round to the nearest cent.
Select one:
a. $1.30
b. $1.31
c. $1.28
d. $1.29
The cost of the tomato sauce for one potion given the cost per ounce is $1.30.
What is multiplication?Multiplication is the process of determining the production of more than one number. It is an example of a mathematical operation.
What is the cost of one portion of the sauce?In order to determine the answer, multiply the cost per ounce by the total ounces needed for one portion.
$0.37 x 3.5 = $1.30
To learn more about multiplication, please check: https://brainly.com/question/13814687
NEED THIS ASAP HELPPP
Answer:
Step-by-step explanation:
The answer is D 28 degrees
Solve for 2. Round to the nearest tenth of a degree, if necessary.
D
to
64
с
37
B
PLS HEP
Step-by-step explanation:
sin X = 37/64
sin X = .578
X = sin ^-1 (.578)
X = 35.3°
hope it's correct
Find questions attached.
Show workings.
You can skip question (a)
The value of x is 40°
Answer:
Solution given:
In triangle BCE
<C+<B+<E=180°[sum of interior angle of a triangle]
<C+60°+x°=180°
<C=180°-60°-x°
<C=120°-x°
again
<A=<C[exterior angle of a cyclic quadrilateral is equal to the opposite interior angle]
2x=120-x
2x+x=120°
3x°=120°
x°=120°/3°
x°=40°
Now we have to,
find the required value of x.
Then in ∆ BCE,
By sum of interior angle of a triangle.
→ <C+<B+<E = 180°
→ <C+60°+x = 180°
→ <C = 180-60-x
→ <C = 120-x
Now by the exterior angle of,
A cyclic quadrilateral is equal to the opposite interior angle.
→ <A = <C
→ 2x = 120-x
→ 2x+x = 120
→ 3x = 120
→ x = 120/3
→ x = 40°
Hence, 40° is the value of x.
