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3 years ago

Rewrite 1−2 sin ^2 (22.5∘) using a double-angle identity.

1 answer:

3 0

Using the identity :

Cos (2a) = 1-2 Sin^2(a)

Therefore :<span>1−2 sin ^2 (22.5∘) = Cos(2a)
= Cos (2 * 22.5) = Cos 45</span>

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Determine whether the given vectors are orthogonal, parallel or neither. (a) u=[-3,9,6], v=[4,-12,-8,], (b) u=[1,-1,2] v=[2,-1,1

nevsk [136]

Answer:

a) $u v= (-3)*(4) + (9)*(-12)+ (6)*(-8)=-168$

Since the dot product is not equal to zero then the two vectors are not orthogonal.

$|u|= \sqrt{(-3)^2 +(9)^2 +(6)^2}=\sqrt{126}$

$|v| =\sqrt{(4)^2 +(-12)^2 +(-8)^2}=\sqrt{224}$

$cos \theta = \frac{uv}{|u| |v|}$

$\theta = cos^{-1} (\frac{uv}{|u| |v|})$

If we replace we got:

$\theta = cos^{-1} (\frac{-168}{\sqrt{126} \sqrt{224}})=cos^{-1} (-1) = \pi$

Since the angle between the two vectors is 180 degrees we can conclude that are parallel

b) $u v= (1)*(2) + (-1)*(-1)+ (2)*(1)=5$

$|u|= \sqrt{(1)^2 +(-1)^2 +(2)^2}=\sqrt{6}$

$|v| =\sqrt{(2)^2 +(-1)^2 +(1)^2}=\sqrt{6}$

$cos \theta = \frac{uv}{|u| |v|}$

$\theta = cos^{-1} (\frac{uv}{|u| |v|})$

$\theta = cos^{-1} (\frac{5}{\sqrt{6} \sqrt{6}})=cos^{-1} (\frac{5}{6}) = 33.557$

Since the angle between the two vectors is not 0 or 180 degrees we can conclude that are either.

c) $u v= (a)*(-b) + (b)*(a)+ (c)*(0)=-ab +ba +0 = -ab+ab =0$

Since the dot product is equal to zero then the two vectors are orthogonal.

Step-by-step explanation:

For each case first we need to calculate the dot product of the vectors, and after this if the dot product is not equal to 0 we can calculate the angle between the two vectors in order to see if there are parallel or not.

Part a

u=[-3,9,6], v=[4,-12,-8,]

The dot product on this case is:

$u v= (-3)*(4) + (9)*(-12)+ (6)*(-8)=-168$

Since the dot product is not equal to zero then the two vectors are not orthogonal.

Now we can calculate the magnitude of each vector like this:

$|u|= \sqrt{(-3)^2 +(9)^2 +(6)^2}=\sqrt{126}$

$|v| =\sqrt{(4)^2 +(-12)^2 +(-8)^2}=\sqrt{224}$

And finally we can calculate the angle between the vectors like this:

$cos \theta = \frac{uv}{|u| |v|}$

And the angle is given by:

$\theta = cos^{-1} (\frac{uv}{|u| |v|})$

If we replace we got:

$\theta = cos^{-1} (\frac{-168}{\sqrt{126} \sqrt{224}})=cos^{-1} (-1) = \pi$

Since the angle between the two vectors is 180 degrees we can conclude that are parallel

Part b

u=[1,-1,2] v=[2,-1,1]

The dot product on this case is:

$u v= (1)*(2) + (-1)*(-1)+ (2)*(1)=5$

Since the dot product is not equal to zero then the two vectors are not orthogonal.

Now we can calculate the magnitude of each vector like this:

$|u|= \sqrt{(1)^2 +(-1)^2 +(2)^2}=\sqrt{6}$

$|v| =\sqrt{(2)^2 +(-1)^2 +(1)^2}=\sqrt{6}$

And finally we can calculate the angle between the vectors like this:

$cos \theta = \frac{uv}{|u| |v|}$

And the angle is given by:

$\theta = cos^{-1} (\frac{uv}{|u| |v|})$

If we replace we got:

$\theta = cos^{-1} (\frac{5}{\sqrt{6} \sqrt{6}})=cos^{-1} (\frac{5}{6}) = 33.557$

Since the angle between the two vectors is not 0 or 180 degrees we can conclude that are either.

Part c

u=[a,b,c] v=[-b,a,0]

The dot product on this case is:

$u v= (a)*(-b) + (b)*(a)+ (c)*(0)=-ab +ba +0 = -ab+ab =0$

Since the dot product is equal to zero then the two vectors are orthogonal.

5 0

3 years ago

Read 2 more answers

What is the range of the following exponential function?plz help asap

IrinaVladis [17]

Answer:

0.23

Step-by-step explanation:

5 0

2 years ago

Please solve, thank you (and show steps):<br><br>Solve for x:<br><br>5x + 22 = 27

IrinaVladis [17]

Given the equation, 5x + 22 = 27:

The goal is to isolate the variable. Hence, a few mathematical operations are necessary to solve for the value of x.

Start by subtracting 22 both sides:
5x + 22 - 22 = 27 - 22
5x + 0 = 5
5x = 5

Next, divide both sides by 5 to isolate and solve for the value of x:

5x/5 = 5/5

x = 1

In order to verify whether x = 1 satisfies the give equation, we must substitute its value into the given equation:

5x + 22 = 27

5(1) + 22 = 27
5 + 22 = 27
27 = 27 (True statement. This implies that x = 1 is the correct value that satisfies the equation).

Therefore, the correct answer is x = 1.

6 0

2 years ago

Read 2 more answers

To stay properly hydrated a person should drink 32 fluid ounces of water every 60 minutes of exercise how much water should Damr

pychu [463]

Write and solve an equation of ratios, as follows:

32 fl oz 60 min
----------- = ------------
x 135 min

Then 60x = (32)(135) = 4320, and x = 72 oz.

damron should drink 72 fl. oz. of water if he rides 135 min.

4 0

3 years ago

An equation for the depreciation of a car is given by y = A(1 – r)t , where y = current value of the car, A = original cost, r =

Mamont248 [21]

<h2>The car is about 6.6 years old.</h2>

Step-by-step explanation:

Given : An equation for the depreciation of a car is given by $y = A(1-r)^t$ , where y = current value of the car, A = original cost, r = rate of depreciation, and t = time, in years. The value of a car is half what it originally cost. The rate of depreciation is 10%.