1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
pashok25 [27]
3 years ago
11

Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) 5, 1,

4
Mathematics
1 answer:
Dahasolnce [82]3 years ago
3 0

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.

You might be interested in
Toby writes an exponential function f(x) that meets the following two conditions:
Mazyrski [523]

II.  f(x) doubles for each increase of 1 in the x values.  Thus, r must be 2, and so we our ar^1 = 6 from ( I ) above becomes  f(x) = a*2^x.  Applying the restriction ar^1 = 6 results in f(1) = a*2^1 = 6, or a = 3.

Then f(x) = ar^x becomes f(x) = 3*2^2 (Answer A)

5 0
3 years ago
Is The ratio 5 gallons in 10 minutes a rate?
andreev551 [17]
It can't be 5gallons In 10 minutes because gallons is physical time not!
8 0
3 years ago
Read 2 more answers
Number 25 please help
8_murik_8 [283]
I believe this is how you solve it

7 0
3 years ago
please help me with my sisters homework answers all of the questions please and show work I need it for today thanks
yawa3891 [41]
For 16 it's 8/10 for 17 it's 52/100 for 18 it's -92/100 for 19 it's -48/100 for 20 it's 86/100 for 21 it's 76/100
6 0
3 years ago
Prove that:- (1-sinA)(1+sinA)÷(1+cosA)(1-cosA)=cot^2A​
astraxan [27]

Answer:

see explanation

Step-by-step explanation:

Using the trigonometric identity

sin²x + cos²x = 1 , then

sin²x = 1 - cos²x and cos²x = 1 - sin²x

Consider the left side

\frac{(1-sinA)(1 + sinA)}{(1+cosA)(1-cosA)} ← expand numerator/denominator using FOIL

= \frac{1-sin^2A}{1-cos^2A}

= \frac{1-(1-cos^2A)}{1-(1-sin^2A)}

= \frac{1-1+cos^2A}{1-1+sin^2A}

= \frac{cos^2A}{sin^2A}

= cot²A = right side , thus proven

6 0
3 years ago
Other questions:
  • If the side lengths of a cube are 16 feet how do you write it in exponential form
    7·1 answer
  • Two horses are ready to return to their barn after a long workout session at the track. the horses are at coordinates H(1,10) an
    6·1 answer
  • Solve the inequality 4x+3&lt;3x+6
    8·1 answer
  • Evaluate f(x) = -x^2 +1 for x=-3
    12·1 answer
  • Please answer the question from the attachment.
    5·1 answer
  • Combine these to like terms to create an equivalent expression: 4z - (-3z)
    13·1 answer
  • A rectangular yard measuring 35ft and 45ft is bordered (and surrounded) by a fence. Inside, a walk that is 4ft wide goes all the
    11·1 answer
  • sapling, or young tree, is planted in a garden. After 3 years, it is 180 cm tall. After 7 years, it is 368 cm tall. How tall wil
    13·1 answer
  • If a circle has a diameter of 21 inches, what is the circumference? (Use 22/7 for π.)
    8·2 answers
  • Find the next 3 terms of each sequence<br> 1. [4, 8, 12, 16 ...}
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!