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ser-zykov [4K]
3 years ago
8

This is abt triangles

Mathematics
1 answer:
Sliva [168]3 years ago
7 0
The answer for this is aaaaaaaaa
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Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) 5, 1,
Dahasolnce [82]

Answer:

The direction cosines are:

\frac{5}{\sqrt{42} }, \frac{1}{\sqrt{42} }  and  \frac{4}{\sqrt{42} }  with respect to the x, y and z axes respectively.

The direction angles are:

40°,  81° and  52° with respect to the x, y and z axes respectively.

Step-by-step explanation:

For a given vector a = ai + aj + ak, its direction cosines are the cosines of the angles which it makes with the x, y and z axes.

If a makes angles α, β, and γ (which are the direction angles) with the x, y and z axes respectively, then its direction cosines are: cos α, cos β and cos γ in the x, y and z axes respectively.

Where;

cos α = \frac{a . i}{|a| . |i|}               ---------------------(i)

cos β = \frac{a.j}{|a||j|}               ---------------------(ii)

cos γ = \frac{a.k}{|a|.|k|}             ----------------------(iii)

<em>And from these we can get the direction angles as follows;</em>

α =  cos⁻¹ ( \frac{a . i}{|a| . |i|} )

β = cos⁻¹ ( \frac{a.j}{|a||j|} )

γ = cos⁻¹ ( \frac{a.k}{|a|.|k|} )

Now to the question:

Let the given vector be

a = 5i + j + 4k

a . i =  (5i + j + 4k) . (i)

a . i = 5         [a.i <em>is just the x component of the vector</em>]

a . j = 1            [<em>the y component of the vector</em>]

a . k = 4          [<em>the z component of the vector</em>]

<em>Also</em>

|a|. |i| = |a|. |j| = |a|. |k| = |a|           [since |i| = |j| = |k| = 1]

|a| = \sqrt{5^2 + 1^2 + 4^2}

|a| = \sqrt{25 + 1 + 16}

|a| = \sqrt{42}

Now substitute these values into equations (i) - (iii) to get the direction cosines. i.e

cos α = \frac{5}{\sqrt{42} }

cos β =  \frac{1}{\sqrt{42} }              

cos γ =  \frac{4}{\sqrt{42} }

From the value, now find the direction angles as follows;

α =  cos⁻¹ ( \frac{a . i}{|a| . |i|} )

α =  cos⁻¹ ( \frac{5}{\sqrt{42} } )

α =  cos⁻¹ (\frac{5}{6.481} )

α =  cos⁻¹ (0.7715)

α = 39.51

α = 40°

β = cos⁻¹ ( \frac{a.j}{|a||j|} )

β = cos⁻¹ ( \frac{1}{\sqrt{42} } )

β = cos⁻¹ ( \frac{1}{6.481 } )

β = cos⁻¹ ( 0.1543 )

β = 81.12

β = 81°

γ = cos⁻¹ ( \frac{a.k}{|a|.|k|} )

γ = cos⁻¹ (\frac{4}{\sqrt{42} })

γ = cos⁻¹ (\frac{4}{6.481})

γ = cos⁻¹ (0.6172)

γ = 51.89

γ = 52°

<u>Conclusion:</u>

The direction cosines are:

\frac{5}{\sqrt{42} }, \frac{1}{\sqrt{42} }  and  \frac{4}{\sqrt{42} }  with respect to the x, y and z axes respectively.

The direction angles are:

40°,  81° and  52° with respect to the x, y and z axes respectively.

3 0
3 years ago
<img src="https://tex.z-dn.net/?f=2%20-%206%20%5Cdiv%206" id="TexFormula1" title="2 - 6 \div 6" alt="2 - 6 \div 6" align="absmid
Karolina [17]

2 - 6 ÷ 6 (Order of Operations - PEMDAS)

= 2 - 1

= 1

Answer is 1

7 0
3 years ago
Toyota campy sold 20,600 in 2008. percent change is 1.2%. how many sold in 2009?
Artyom0805 [142]
So 20600=100%
in 2009=1.2% change from 100%

not specific whether percent change is up or down so solve fo r1.2%
d
find 1.2% of 100
percent means parts out of 1001.2%=1.2/100=0.12/10=0.012/1=0.012

'of' means multiply
20600 times 0.012=276.2

the change is 276.2

20600-276.2=20352.8
20600+276.2=20847.2
you must round down since you can't sell 0.2 or 0.8 of a car

the sold either 20,352 or 20,847 cars in 2009

6 0
3 years ago
Petrolyn motor oil is a combination of natural oil and synthetic oil. It contains 5 liters of natural oil for every 8 liters of
Sloan [31]

9514 1404 393

Answer:

  340

Step-by-step explanation:

The ratio of natural to synthetic is give as ...

  5 : 8

Then the ratio of natural to total would be ...

  5 : (5+8) = 5 : 13

The amount of natural oil in 884 total liters of oil will be ...

  (5/13)(884 L) = 340 L

884 liters of Petrolyn oil will contain 340 liters of natural oil.

4 0
2 years ago
Read 2 more answers
The sum of two numbers is 19. The difference is 9. What are the two numbers?
Nostrana [21]

Answer:

the two numbers are 8 and 8

7 0
3 years ago
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