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ryzh [129]
3 years ago
12

Determine if the two vectors u = –3i – j and v = 2i + 2j are orthogonal. If they are not orthogonal, specify the angle between t

hem
Mathematics
1 answer:
NemiM [27]3 years ago
3 0
Two vectors by definition are orthogonal when the angle between them is 90 degrees.
 It can also be said that they are orthogonal if the scalar product between both vectors is zero.
 We have then:
 u = -3i - j
 v = 2i + 2j
 u.v = (- 3i - j) (2i + 2j) = - 6-2
 u.v = -8
 They are not orthogonal vectors.
 The angle between them is
 lul = root ((- 3) ^ 2 + (-1) ^ 2) = 3.16227766
 lvl = root ((2) ^ 2 + (2) ^ 2) = 2.828427125
 The angle between them is:
 cos (x) = (u.v) / ((lul) * (lvl))
 cos (x) = (- 8) / ((3.16227766) * (2.828427125))
 cos (x) = - 0.894427191
 x = acos (-0.894427191)
 x = 153.4349488
 x = 153.43
 Answer: 
 It is not orthogonal
 Angle between them: 
 x = 153.43
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Answer:

a) y=0.00991 x +1.042  

b) r^2 = 0.7503^2 = 0.563

c) r=\frac{7(30095)-(4210)(49)}{\sqrt{[7(2595100) -(4210)^2][7(354) -(49)^2]}}=0.7503  

Step-by-step explanation:

Data given

x: 500, 700, 750, 590 , 540, 650, 480

y: 7.00, 7.50 , 9.00, 6.5, 7.50 , 7.0, 4.50

Part a

We want to create a linear model like this :

y = mx +b

Wehre

m=\frac{S_{xy}}{S_{xx}}  

And:  

S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}  

S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}  

With these we can find the sums:  

S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=2595100-\frac{4210^2}{7}=63085.714  

S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i){n}}=30095-\frac{4210*49}{7}=625  

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Nowe we can find the means for x and y like this:  

\bar x= \frac{\sum x_i}{n}=\frac{4210}{7}=601.429  

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And we can find the intercept using this:  

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Part b

The correlation coefficient is given by:

r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}  

For our case we have this:

n=7 \sum x = 4210, \sum y = 49, \sum xy = 30095, \sum x^2 =2595100, \sum y^2 =354  

r=\frac{7(30095)-(4210)(49)}{\sqrt{[7(2595100) -(4210)^2][7(354) -(49)^2]}}=0.7503  

The determination coefficient is given by:

r^2 = 0.7503^2 = 0.563

Part c

r=\frac{7(30095)-(4210)(49)}{\sqrt{[7(2595100) -(4210)^2][7(354) -(49)^2]}}=0.7503  

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