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Vectors and Coordinate Geometry

Vectors and Coordinate Geometry

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Directional Cosine


 »  Directional cosines:

    →  `vec p = ai+bj+ck` makes angles `alpha,beta,gamma` with `x,y,z`-axes respectively

    →  `cos alpha= a/(|p|)`

    →  `cos beta= b/(|p|)`

    →  `cos gamma= c/(|p|)`

Directional Cosine

plain and simple summary

nub

plain and simple summary

nub

dummy

Directional cosines are the ratio of projections on to an axes to the magnitude.

simple steps to build the foundation

trek

simple steps to build the foundation

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In this page, you will learn about the alternate representation of vectors other than `ai+bj+ck`?


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Starting on learning "Directional Cosine". ;; In this page, you will learn about the alternate representation of vectors other than a i + b j + c k ?

What is 'cosine ' of an angle?

  • Cosine of an angle does not exist.
  • the ratio of adjacent side to the hypotenuse in a right angled triangle

Answer is 'the ratio of adjacent side to the hypotenuse in a right angled triangle'.

While referring to vector quantities, we used to switch between two representations. One with components along the axes. What is the other?

  • magnitude
  • angles with the axes
  • magnitude along with angles with the axes
  • none of the above

Answer is 'magnitude along with angles with the axes'. An alternate representation to component form is to specify magnitude and angles made by the vector on the axes.

What information is complete to describe the vector `ai+bj` shown in the figure?directional cosine of 2D vector

  • `a`
  • `r`
  • `theta`
  • `r` and `theta`

answer is '`r` and `theta`'. If these two parameters are available we can derive the vector notation `ai+bj`.

What is the value of `a` and `b` in the figure?example of directional cosine of 2D vector

  • `a=8cos60` and `b=8sin60`
  • `a=8` and `b=8`
  • `a=cos60` and `b=sin60`
  • `a=sqrt(8^2cos60)` and `b=sqrt(8^2sin60)`

answer is '`a=8 cos 60` and `b=8 sin 60`'.

What information is complete to describe the vector `ai+bj+ck` shown in figure?directional cosine of 3D vector

  • `a`
  • only `r`
  • `alpha, beta, gamma`
  • `r` and `alpha, beta`

answer is '`r` and `alpha, beta`'. If these three parameters are available we can derive the vector notation `ai+bj+ck`.

Note that the third angle can be derived from the other two angles.

What are the values of `a`, `b`, and `c` in the figure?example of directional cosine of 3D vector

  • `a=3cos60`; `b=3cos45`; `c=3cos75`
  • `a=3`; `b=3`; `c=3`
  • `a=cos60`; `b=cos45`; `c=cos75`
  • `a=60`; `b=45`; `c=75`

answer is '`a=3cos60`; `b=3cos45`; `c=3cos75`'

When a vector with magnitude `r` and angles `alpha,beta,gamma` is given, the coordinate form of the vector is
`r(cos alpha i+cos beta j+cos gamma k)`
`cos alpha`, `cos beta`, `cos gamma` are called the directional cosines of the vector.

The mathematical representation of vectors is the component form `vec p = ai+bj+ck`
Directional cosines along with magnitude provide an alternate representation of a vector.
`vec p = r (cos alpha i + cos beta j + cos gamma k)`

What is the magnitude of the directional cosines vector : `cos alpha i+cos beta j+cos gamma k` ?

Note that

`cos alpha =a/r`

`cos beta =b/r`

`cos gamma =c/r`

`r^2=a^2+b^2+c^2`

  • `1`
  • `0`
  • `r`
  • `sqrt(r)`

Answer is '`1`'. The magnitude = `sqrt(cos^2 alpha+cos^2 beta+cos^2 gamma)`, which evaluates to 1.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Directional cosines: Given that a vector `vec p = ai+bj+ck` makes angles `alpha,beta,gamma` with `x,y,z`-axes respectively, then the directional cosines of the vector are
`cos alpha= a/(|p|)`
`cos beta= b/(|p|)`
`cos gamma= c/(|p|)`

Directional cosines make Unit Vector: For a given vector `ai+bj+ck` the directional cosine vector `li+mj+nk` is the unit vector in the direction of the given vector. Note that

`l = cos alpha =a/r`

`m = cos beta =b/r`

`n = cos gamma =c/r`
This also implies that

`l^2+m^2+n^2=1`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Find the direction cosines of a line which makes equal angles with the coordinate axes.

  • tap for the answer

Answer is
`(1/sqrt(3),1/sqrt(3),1/sqrt(3))` OR `(-1/sqrt(3),-1/sqrt(3),-1/sqrt(3))`

From the question, we know that `alpha=beta=gamma` and We know the property of directional cosines `cos^2 alpha+cos^2 beta+cos^2 gamma = 1 `

If a line makes angles `45,135, 90` with the x, y and z-axes respectively, find its direction cosines.

  • tap for the answer

Answer is '`cos45, cos135, cos90`'.

If a line makes angles `30,60,90` with the x, y and z-axes respectively, find its direction cosines.

  • tap for the answer

Answer is '`cos30, cos60, cos90`'

Find the directional cosine of the vector `2i-3j+sqrt(3)k`.

  • tap for the answer

magnitude of the vector = 4
So directional cosines are `2/4 , -3/4, sqrt(3)/4`

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Progress

What is 'cosine ' of an angle?
cosine;angle;exist
Cosine of an angle does not exist.
ratio;adjacent;hypotenuse;right;angled;triangle
the ratio of adjacent side to the hypotenuse in a right angled triangle
Answer is 'the ratio of adjacent side to the hypotenuse in a right angled triangle'.
While referring to vector quantities, we used to switch between two representations. One with components along the axes. What is the other?
magnitude
magnitude
angles
angles with the axes
along with;
magnitude along with angles with the axes
none;above
none of the above
Answer is 'magnitude along with angles with the axes'. An alternate representation to component form is to specify magnitude and angles made by the vector on the axes.
What information is complete to describe the vector, a i + b j, shown in the figure?
a
a
r
r
theta
theta
and
r and theta
The answer is "r and theta". If these two parameters are available we can derive the vector notation a i + b j.
What is the value of a and b in the figure?
1
2
3
4
The answer is "a = 8 cos 60 and b = 8 sine 60".
What information is complete to describe the vector, a i + b j + c k, shown in figure?
a
a
only
only r
gamma
alpha, beta, gamma
r and; and alpha
r and alpha, beta
The answer is "r and alpha beta". If these three parameters are available we can derive the vector notation a i+b j+c k. ;; Note that the third angle can be derived from the other two angles.
What are the values of a , b , and c in the figure?
1
2
3
4
The answer is "a = 3 cos 60 ; b - 3 cos 45 ; c = 3 cos 75".
When a vector with magnitude r and angles alpha, beta, gamma is given, the coordinate form of the vector is r multiplied (cos alpha i + cos beta j +cos gamma k);; cos alpha, cos beta, cos gamma are called the directional cosines of the vector.
Directional cosines are the ratio of projections on to an axes to the magnitude.
Directional cosines: Given that a vector p = a i + b j + c k , makes angles alpha, beta, gamma with x y z axes respectively, then the directional cosines of the vector are ;; cos alpha = a by magnitude of p ;; cos beta = b by magnitude of p ;; and cos gamma = c / magnitude of p.
The mathematical representation of vectors is the component form; vector p = a i+b j+c k;; Directional cosines along with magnitude provide an alternate representation of a vector. vector p = r (cos alpha i + cos beta j + cos gamma k)
What is the magnitude of the directional cosines vector : cos alpha i + cos beta j + cos gamma k ?
1
1
0
0
r
r
square;root
square root of r
The answer is "1". The magnitude = square root cos squared alpha + cos squared beta + cos squared gamma, which evaluates to 1.
Directional cosines make Unit Vector: For a given vector a i + b j + c k, the directional cosine vector l i + m j + n k is the unit vector in the direction of the given vector. Note that ;; l = cos alpha = a by r ;; m = cos beta = b by r ;; n = cos gamma = c by r ;; This also implies that l squared + m squared + n squared = 1.
Find the direction cosines of a line which makes equal angles with the coordinate axes.
1
The answer is given.
If a line makes angles 45,135, 90 with the x, y and z-axes respectively, find its direction cosines.
1
The answer is "cos 45, cos 135, cos 90"
If a line makes angles 30,60,90 with the x, y and z-axes respectively, find its direction cosines.
1
The answer is "cos 30, cos 60, cos 90"
Find the directional cosine of the vector 2 i minus 3 j + square root 3 k
1
The answer is "2 by 4, minus 3 by 4, square root of 3 by 4 ".

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