You look for the same variables with the same powers
y: 16y +8y = 24y
x: 8x - 7x = x
y^2: 2y^2
and sum it all together at the end
In a symmetric distribution, better known as the Gaussian or normal distribution, two thirds of the observations or 66.67% of the observation belongs to the area between the -3∅ to +3∅ where ∅ is the standard deviation of the group. it is assumed here that the mean is 1 and the area under the curve is also 1.
Answer:
The probability that both balls are red is 5.81%, that one is red and one black is 38.36%, and that both are black 55.81%.
Step-by-step explanation:
Given that a box contains 44 table tennis balls having the same shape and size, and among them 33 are black and remaining balls are red, if two balls are drawn randomly one after another without replacement, to show the probability of all possible outcomes the following calculations must be done:
Red + red:
11/44 + 10/43 = 0.25 x 0.2325 = 0.0581
Red + black:
11/44 + 33/43 = 0.25 x 0.7674 = 0.1918
Black + red:
33/44 + 11/43 = 0.75 x 0.2558 = 0.1918
Black + black:
33/44 + 32/43 = 0.75 x 0.7441 = 0.5581
19.18 x 2 = 38.36
Therefore, the probability that both balls are red is 5.81%, that one is red and one black is 38.36%, and that both are black 55.81%.
<span>1. Find a ⋅ b.
a = <7, 4>, b = <3, 5> (2 points)
41 <---------- right answer
<10, 9>
-1
<21, 20>
Explanation:
a . b means the dot or scalar product of the vectors a and b.
The scalar product of vectorr <x1,y2> and <x2,y2> is (x1)*(y1) + (x2)*(y2)
So, a . b = 7*3 + 4*5 = 21 + 20 = 41 <-------- answer
2. Find a ⋅ b
a = 10i + 9j, b = 4i + 3j (2 points)
<40, 27>
<14, 12>
67 <---------- answer
-13
Explanation:
Again you are required to find the dot product of two vectors. In this case the vectors are given using the representation with the unit vectors i and j.
The dot product of the vectors (x1 i + y1 j) . (x2 i + y2 j) is x1*x2 + y1*y2
So, (</span><span><span>10i + 9j) . (4i + 3j) = 10*4 + 9*3 = 40 + 27 = 67
Answer: 67
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3. Find the angle between the given vectors to the nearest tenth of a degree.
u = <6, -1>, v = <7, -4> (2 points)
20.3° <--------------- answer
10.2°
0.2°
30.3°
Explanation:
You can use the dot product to find the angle between two vectors.
This is the formula:
cos(α) = [ <a> dot <b. ] / [|a| |b| ]
where α is the angle between the vector a and b.
<a> is the vector a, <b> is the vector b, <a> dot <b> is the dot product, |a| is the magnitude of vector a, and |b| is the magnitude of vector b.
<u> dot <v> = (6)(7) + (-1)(-4) = 42 + 4 = 46
|u| = √ (6^2 + 1^2) = √37
|v| = √ (7^2 + 4^2) = √65
=> cos(α) = [46] / [√37 * √65] = 0.938
=> α = arccos(0.938) = 20.3°
4. Determine whether the vectors u and v are parallel, orthogonal, or neither.
u = <10, 6>, v = <9, 5> (2 points)
Orthogonal
Parallel
Neither <------------ answer
Explanation:
At sight they are neither orthogonal nor parallel. You can put the points in a graph and you will realize inmediately.
Let's calculate the angle, with the formula given in the above problem.
cos(α) = [<u> dot< >v] / [ |u| |v| ]
<u> dot <v> = 10*9 + 6*5 = 90 + 30 = 120
|u| = √(10^2 + 6^2) = √136
|v| = √(9^2 + 5^2) = √106
cos(α) = 120 / (√136 * √106] = 0,999
α = arctan(0,999) = 1,9°
Which confirms that they are neither orthogonal nor parallel
5. Evaluate the expression:
v ⋅ w
Given the vectors:
r = <8, 8, -6>; v = <3, -8, -3>; w = <-4, -2, -6> (2 points)
</span>
Solution:
v . w = v dot w = (3)(-4) + (-8)(-2) + (-3)(-6) = -12 + 16 + 18 = 22
Answer: 22