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»  Components in parallel add and component in perpendicular combine as right triangle
→  component in parallel to vec p is vec a
→  component in perpendicuar to vec p is vec b
→  vec p + vec a and vec b result in vec r

### Vector Addition - First principles

plain and simple summary

nub

plain and simple summary

nub

dummy

•  the components in parallel (in the same direction) are directly added in magnitude and
•  the components in perpendicular are combined using Pythagoras theorem.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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Starting on learning "Vector addition explained in first principles". ;; In this page, you will learn first principles of how two vectors quantities add up?

So far, we understood
•  vectors are quantities with magnitude and direction
•  vectors are represented in the a form that includes the direction, eg: component form ai+bj+ck

Can you identify the vector quantity in the following list?

• 3 meter north
• 3 meter
• north
• none of the above

Answer is '3 meter north' which specifies magnitude 3 and direction 'north'.

A person walks 3 meter north and continues in the same direction for another 4 meter. What is the distance he has to walk to return back to the starting position?

• =3+4
• !=3+4

Since the person walks towards north all the time, the answer is '3+4'.
Two vector quantities add up in magnitude if they are in same direction.

A person walks 3 meter north and continues towards east for another 4 meter. what is the distance he has to walk to return back to the starting position?

• =3+4
• !=3+4

Since the person walks in different directions, the answer: '!=3+4'. Two vector quantities do not add up in magnitude if they are not in same direction. The result magnitude will be smaller to the magnitudes of the vectors being added.

A person walks 3 meter north-east and continues to walk in north-east direction for another 4 meter. what is the distance he has to walk to return back to the starting position?

• =3+4
• !=3+4

Since the person walks in only one direction, the answer is '3+4'. Two vector quantities add up in magnitude if they are in same direction.

A person walks 3 meter north-east. He then turns back towards the starting point and walks 2 meter towards starting point. Then, what is the distance he has to walk to return back to the starting position?

• =3+2
• =3-2
• !=3-2

The second stretch of walk is considered to be -2 in the same direction as the first stretch. One vector subtract from the another vector in magnitude if they are in opposite directions.

What we have learned in these examples?

• the magnitudes can be added if the vectors are in the same direction
• the magnitudes can not be added if the vectors are in the same direction
• the magnitudes can be added irrespective of the direction of the vectors
• the magnitudes cannot be added regardless of any information of direction

Answer is, 'the magnitudes can be added if the vectors are in the same direction'.

A vector of length 2 unit at angle 24^@ is added with another vector of 2.3 unit at angle 24^@. What is the resulting vector?

• 2.3 unit at angle 24^@
• 4.3 unit scalar
• 4.3 unit at angle 24^@
• 4.3 unit at angle 48^@

Answer is '4.3 unit at angle 24^@'. Since the direction is same, the vectors add up in magnitude.

A vector of length 1 unit at angle 24^@ is followed by another vector of 2.3 unit at angle 43^@. What is the resulting vector?

• The magnitudes add up 1+2.3
• The directional angles add up 24^@+43^@
• The magnitudes add up to a value less than 1+2.3
• none of the above

Answer is 'The magnitudes add up to a value less than 1+2.3'.

A vector of length 1 unit at angle 24^@ is followed by another vector of 2.3 unit at angle 43^@.
The problem is illustrated in figure. How this can be solved to find the result?

• use trigonometrical calculations
• cannot use trigonometrical calculations

The trigonometrical calculations help in solving this problem.

If the vector in the question represents one of
•  force
•  electric field
•  velocity
•  displacement

. And we have a coordinate geometry representation of lines in the figure. Which of the vector quantities can be equivalently represented in a coordinate plane and solved using geometry?

• All forms of vectors
• Only distances represented as vectors
• none of the vectors

Answer is 'all forms of vectors can be equivalently represented in coordinate plane as rays and geometry can be used to solve problems'

The vector of 1 unit at 24^@ is considered as hypotenuse of right angle triangle. The sides are calculated as cos24^@ and sin24^@. The same for the vector 2.3 unit at angle 43^@, the sides are calculated as 2.3cos43^@ and 2.3sin43^@. This two can be used to find the sides for the result vec(OP).

what is vec(OP)?

• cos24^@+2.3cos43^@
• sin24^@+2.3sin43^@
• (cos24^@+2.3cos43^@)+(sin24^@+2.3sin43^@)
• (cos24^@+2.3cos43^@)i+(sin24^@+2.3sin43^@)j

Answer is '(cos24^@+2.3cos43@)i+(sin24^@+2.3sin43^@)j' as evident from the figure.

Summary:
If two vectors are added, then the result of addition is to be computed using trigonometrical calculations.

If the quantities have the same direction, then the trigonometrical calculations are simple enough that their magnitudes add up.

If the quantities have directions at right angle, then Pythagoras theorem can be used to combine the magnitudes.

If the quantities have different directions, then trigonometric calculations are used to find components in parallel and in perpendicular.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Vector Addition First Principles: When two vectors vec p and vec q are added, vector vec q is split into

•  component vec a in parallel to vec p and

•  component vec b in perpendicular to vec p

vec a is combined to the vec p and vec b is combined using trigonometry.

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Progress

Progress

So far, we understood ;; vectors are quantities with magnitude and direction ;; vectors are represented in the a form that includes the direction, eg: component form a i+b j+c k
Can you identify the vector quantity in the following list?
1
2
3
4
Answer is ' 3 meter north' which specifies magnitude 3 and direction 'north'.
A person walks 3 meter north and continues in the same direction for another 4 meter. What is the distance he has to walk to return back to the starting position?
1
2
Since the person walks towards north all the time, the answer is ' 3+4 '. ;; Two vector quantities add up in magnitude if they are in same direction.
A person walks 3 meter north and continues towards east for another 4 meter. what is the distance he has to walk to return back to the starting position?
1
2
Since the person walks in different directions, the answer: not equal 3+4 . ;; Two vector quantities do not add up in magnitude if they are not in same direction. ;; The result magnitude will be smaller to the magnitudes of the vectors being added.
A person walks 3 meter north-east and continues to walk in north-east direction for another 4 meter. What is the distance he has to walk to return back to the starting position?
1
2
Since the person walks in only one direction, the answer is ' 3+4 '. ;; Two vector quantities add up in magnitude if they are in same direction.
A person walks 3 meter north-east. He then turns back towards the starting point and walks 2 meter towards starting point. Then, what is the distance he has to walk to return back to the starting position?
1
2
3
The second stretch of the walk is considered to be -2 in the same direction as the first stretch. ;; One vector subtract from the another vector in magnitude if they are in opposite directions.
What we have learned in these examples?
1
2
3
4
Answer is, 'the magnitudes can be added if the vectors are in the same direction'.
A vector of length 2 unit at angle 24 degree is added with another vector of 2 point 3 unit at angle 24 degree. What is the resulting vector?
1
2
3
4
Answer is '4 point 3 unit at angle 24 degree'. Since the direction is same, the vectors add up in magnitude.
A vector of length 1 unit at angle 24 degree is followed by another vector of 2 point 3 unit at angle 43 degree. What is the resulting vector?
1
2
3
4
Answer is: The magnitudes add up to a value less than 1+2 point 3
A vector of length 1 unit at angle 24 degree is followed by another vector of 2 point 3 unit at angle 43 degree. The problem is illustrated in figure. How this can be solved to find the result?
1
2
The trigonometrical calculations help in solving this problem.
If the vector in the question represents one of ;; force ;; electric field ;; velocity ;; displacement. And we have a coordinate geometry representation of lines in the figure. Which of the vector quantities can be equivalently represented in a coordinate plane and solved using geometry?
all;forms
All forms of vectors
only;distances
Only distances represented as vectors
none
none of the vectors
Answer is 'all forms of vectors can be equivalently represented in coordinate plane as rays and geometry can be used to solve problems'
The vector of 1 unit at 24 degree is considered as hypotenuse of right angle triangle. The sides are calculated as cos24 and sin24 . The same for the vector 2 point 3 unit at angle 43 degree, the sides are calculated as 2point 3cos43 and 2 point 3sin43 . This two can be used to find the sides for the result vector o p. what is vector o p ?
1
2
3
4
The answer is "cos 24 + 2 point 3 cos 43 i + sin 24 + 2 point 3 sin 43" as evident from the figure.
Summary: If two vectors are added, then the result of addition is to be computed using trigonometrical calculations. ;; If the quantities have the same direction, then the trigonometrical calculations are simple enough that their magnitudes add up. ;; If the quantities have directions at right angle, then Pythagoras theorem can be used to combine the magnitudes. ;; If the quantities have different directions, then trigonometric calculations are used to find components in parallel and in perpendicular.
When vectors are added, ;; the components in parallel (in the same direction) are directly added in magnitude ;; the components in perpendicular are combined using Pythagoras theorem
Vector Addition First Principles: When two vectors p and q are added, vector q is split into ;; component vector a in parallel to vector p and ;; component vector b in perpendicular to vector p ;; vector a is combined to vector p and vector b is combined using trigonometry.

we are not perfect yet...