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Thought-Process to Discover Knowledge

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

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

User Guide

Welcome to nubtrek.

The content is presented in small-focused learning units to enable you to
think,
figure-out, &
learn.

Just keep tapping (or clicking) on the content to continue in the trail and learn.

User Guide

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

nub,

trek,

jogger,

exercise.

User Guide

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about.

User Guide

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step.

User Guide

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This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen.

User Guide

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge.

summary of this topic

### Vectors and Coordinate Geometry

Voice

Voice

Home

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

trek

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

Keep tapping on the content to continue learning.
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.

(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.

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.

Answer is 'cos30, cos60, cos90'

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

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

Progress

Progress

What is 'cosine ' of an angle?
cosine;angle;exist
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
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