In this page, you will learn first principles of how two vectors quantities add up?

*click on the content to continue..*

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
- `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`
- `!=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`
- `!=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`
- `!=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`
- `!=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 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 `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`
- 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
- 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
- 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`
- `(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.

When vectors are added,

• *the components in parallel (in the same direction) are directly added in magnitude* and

• *the components in perpendicular are combined using Pythagoras theorem*.

**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.

*slide-show version coming soon*