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Thought-Process to Discover Knowledge

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learn.

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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

jogger provides the complete mathematical definition of the concepts.

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

### Direction - Unique Feature of Vectors

Voice

Voice

Home

Vector Arithmetic

»  Application Scenarios of Vector Arithmetic

→  Subtraction : inverse of addition

→  Repeated addition of a quantity: a vector is multiplied by a scalar

→  Product between two quantities : two vectors can be multiplied

»  Two Possibilities of Vector Multiplication

→  Vector Dot Product : multiply with component in parallel

→  Vector Cross Product : multiply with component in perpendicular

»  Vector Division is not defined
→  Division is the inverse of multiplication. In vector multiplication, one of the components is lost in the product. And so vector division is not defined.

All these arithmetic operations are explained in detail later. The objective of this is to understand the role of components in parallel and in perpendicular.

### Vector Arithmetic

plain and simple summary

nub

plain and simple summary

nub

dummy

Vector Arithmetic involves

•  repeated addition or multiplication by scalar

•  two types of vector products

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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In this page, the basic mathematical operations of vector algebra is introduced with examples.

Keep tapping on the content to continue learning.
Starting on learning "Vector Arithmetic". ;; In this page, the basic mathematical operations of vector algebra is introduced with examples.

A person takes 2 apples and then takes 3 oranges. How many fruits the person has?

• 3
• 2
• 5
• none of the above

The answer is '5' fruits. In this scalar quantities 2 and 3 add to the sum 5

A person takes 6 apples and then puts back 4 apples. How many fruits the person has?

• 3
• 2
• 5
• none of the above

The answer is '2' fruits. In this scalar quantity 4 is subtracted from 6 to get result 2.

Scalar quantities can be subtracted

A person takes 6 apples and then takes -4 apples. How many fruits the person has?

• 3
• 2
• 5
• none of the above

The answer is '2' fruits. In this, 6 is added with -4, which is effectively subtraction in the form (6-4).

Subtraction is the inverse of addition.

A person takes 2 apples and places them in his basket. He repeats that 3 times in total. How many apples does the person have in the basket?

• 2+2+2 apples
• 2xx3 apples
• 6 apples
• all the above

Answer is 'all the above'. In this, scalar quantity 2 is repeatedly added 3 times to get result 6.

A scalar quantity can be added repeatedly.
Repeated addition is a form of multiplication.

A person having 10 apples in her basket, divides that equally to two kids. How many apples will a kid have?

• 10-:2
• 5
• count the number of twos in 10
• all the above

Answer is 'all the above'. In this, scalar quantity 10 is divided by 2.

A scalar quantity can be divided.

A person having 10 apples in her basket, gives half of them to a kid. How many apples will the kid have?

• 10xx1/2
• 10-:2
• 5
• all the above

Answer is 'all the above&rsquo. In this 10 is multiplied by half, which is equivalently the division 10-:2.

Division is the inverse of multiplication.

A person takes 3 oranges that costs 10 coins each. How much the person has to pay for the fruits?

• 40 coins
• 30 coins
• 10 coins
• 50 coins

The answer is '30' coins. The scalar quantity 3 multiplies with another scalar quantity 10.

Two different scalar quantities can be multiplied.

Summary

•  Scalar quantities have magnitude measure.

•  Scalar quantities can be added (2 fruit + 3 fruit = 5 fruit)

•  A scalar quantity can be multiplied (repeated addition) (2 apples 3 times = 6 apples)

•  scalar quantities can interact to form a product scalar quantity (3 oranges at 10 coins each = 30 coins)

Apart from the addition and multiplication,

•  scalar quantities can be subtracted (3 fruits - 2 fruits) = 1 fruit; which is inverse of addition

•  Scalar quantities can be divided, which is inverse of multiplication.

Fundamental mathematical operations

•  product of quantities (multiplication)

Let us see How these are defined for vectors that has magnitude and direction.

A person walks 3 meter north and 4 meter east, what is the final distance (in meter) from the starting point?

• 3+4
• 7
• both the above
• sqrt(3^2+4^2)

Answer is 'sqrt(3^2+4^2)'. In this vector quantity 3 meter north and 4 meter east are added. We used coordinate geometry to figure out the result.

A person walks 3m in a direction and continues to do the same four times. What is the final position of him from the starting point?

• 3xx4 meter in the specified direction
• 3+3+3+3 meter in the specified direction
• both the above

Vector quantities can be repeatedly added or equivalently – vector can be multiplied by a scalar quantity.

A person pushes with 3 unit force toward east and causes 2 unit displacement towards north-east. Given that
work =product of force and displacement
Can you make a guess what is the work done by the person?

• force as a vector and displacement as a vector are multiplied
• force as a vector is multiplied by magnitude of displacement
• magnitude of force and magnitude of displacement are multiplied
• none of the above

The answer is, 'force as a vector and displacement as a vector are multiplied'. Over this course, this will be explained in detail.

Vectors can be multiplied.

Apart from the addition, multiplication of vector by a scalar, and vector products,

•  vectors can be subtracted, which is 'inverse of addition' .

•  vectors can be divided by a scalar, which is included as 'inverse of multiplication of a vector by a scalar'

•  Vector division is not defined. We’ll see why in due course of this learning.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Vector Arithmetic: Given vectors vec p and vec q,

•  vector addition: vec p + vec q

•  multiplication of vector by scalar: vec p + vec p + cdots (n-text(times)) = n vec p

•  vector dot product: vec p multiplied by component of vec q in parallel to vec p

•  vector cross product: vec p multiplied by component of vec q in perpendicular to vec p

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Progress

Progress

A person takes 2 apples and then takes 3 oranges. How many fruits the person has?
3
3
2
2
5
5
none;above
none of the above
The answer is "5" fruits. In this scalar quantities 2 and 3 add to the sum 5. ;; Scalar quantities can be added.
A person takes 6 apples and then puts back 4 apples. How many fruits the person has?
3
3
2
2
5
5
none;above
none of the above
The answer is "2" fruits. In this scalar quantity 4 is subtracted from 6 to get result 2. ;; Scalar quantities can be subtracted.
A person takes 6 apples and then takes minus 4 apples. How many fruits the person has?
3
3
2
2
5
5
none;above
none of the above
The answer is "2" fruits. In this, 6 is added with minus 4, which is effectively subtraction in the form 6 minus 4. ;; Subtraction is the inverse of addition.
A person takes 2 apples and places them in his basket. He repeats that 3 times in total. How many apples does the person have in the basket?
plus
2+2+2 apples
times
2 times 3 apples
6
6 apples
all;above
all the above
The answer is "all the above". In this, scalar quantity 2 is repeatedly added 3 times to get result 6. ;; A scalar quantity can be added repeatedly. ;; Repeated addition is a form of multiplication.
A person having 10 apples in her basket, divides that equally to two kids. How many apples will a kid have?
divided
10 divided by 2
5
5
count;number;in;
count the number of twos in 10
all;above
all the above
The answer is "all the above". In this, scalar quantity 10 is divided by 2. A scalar quantity can be divided.
A person having 10 apples in her basket, gives half of them to a kid. How many apples will the kid have?
times
10 times half
divided
10 divided by 2
5
5
all;above
all the above
The answer is "all the above". In this 10 is multiplied by half, which is equivalently the division 10 divided by 2. ;; Division is the inverse of multiplication.
A person takes 3 oranges that costs 10 coins each. How much the person has to pay for the fruits?
40
40 coins
30
30 coins
10
10 coins
50
50 coins
The answer is "30 coins". The scalar quantity 3 multiplies with another scalar quantity 10. ;; Two different scalar quantities can be multiplied.
Summary : Scalar quantities have magnitude measure.;; Scalar quantities can be added ;; A scalar quantity can be multiplied (repeated addition) ;; scalar quantities can interact to form a product scalar quantity.
Apart from the addition and multiplication, ;; scalar quantities can be subtracted; which is inverse of addition ;; Scalar quantities can be divided, which is inverse of multiplication.
Fundamental mathematical operations: addition;; repeated addition (multiplication) ;; product of quantities (multiplication) ;; Let us see How these are defined for vectors that has magnitude and direction.
A person walks 3 meter north and 4 meter east, what is the final distance from the starting point?
3;plus;4
3+4
7
7
both;above
both the above
square;root;
square root 3 squared + 4 squared
The answer is "square root 3 squared + 4 squared". In this vector quantity 3 meter north and 4 meter east are added. We used coordinate geometry to figure out the result. ;; Vector quantities can be added.
A person walks 3 meter in a direction and continues to do the same four times. What is the final position of him from the starting point?
times
3 times 4 meter in the specified direction
plus
3+3+3+3 meter in the specified direction
both;above
both the above
The answer is "both the above". Vector quantities can be repeatedly added or equivalently – vector can be multiplied by a scalar quantity.
A person pushes with 3 unit force toward east and causes 2 unit displacement towards north-east. Given that work equals force multiplied displacement ;; Can you make a guess what is the work done by the person?
1
2
3
4
The answer is " force as a vector and displacement as a vector are multiplied ". Over this course, this will be explained in detail. ;; Vectors can be multiplied.
Vector arithmetic involves ;; addition;; repeated addition or multiplication by scalar;; two types of vector products.
Vector Arithmetic: Given vectors p and q.;; vector addition: vector p + vector q ;; multiplication of vector by scalar: vector p + vector p + n times = n times vector p ;; vector dot product: vector p multiplied by component of vector q in parallel to vec p ;; vector cross product: vector p multiplied by component of vector q in perpendicular to vec p
Apart from the addition, multiplication of vector by a scalar, and vector products, ;; vectors can be subtracted, which is 'inverse of addition'.;; vectors can be divided by a scalar, which is included as 'inverse of multiplication of a vector by a scalar' ;; Vector division is not defined. We’ll see why in due course of this learning.

we are not perfect yet...