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` * Scalar quantities can be added.*

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

• 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 (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. *Vector quantities can be added.*

A person walks `3`m 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

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 `=`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

• Addition

• 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.

*slide-show version coming soon*