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mathsVector AlgebraDirection - Unique Feature of Vectors

### Vector Arithmetic

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

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

• 3
• 2
• 5
• 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
• 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
• 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
• 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
• 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
• 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
• 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)
• 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
• 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 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.

Vector Arithmetic involves

•  repeated addition or multiplication by scalar

•  two types of vector products

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

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.

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