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Thought-Process to Discover Knowledge

Welcome to **nub****trek**.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

Welcome to **nub****trek**.

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The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

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exercise.

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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|)`

*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*

trek

You are learning the free content, however do shake hands with a coffee to show appreciation.

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

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?

- `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?

- `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?

- `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?

- `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`

*your progress details*

Progress

*About you*

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 ".