Vectors are quantities with magnitude and direction.

How are the vectors represented mathematically?

Understanding that is the objective of this session.

*click on the content to continue..*

Identify one quantity with direction in the following.

- `3` meter long
- `3` meter North
- `3` meter North
- `3` meter rope
- `3` meter width

Answer is '`3` meter North' - In this the direction 'North' is given.

If someone is calculating the position of an airplane, what would be a reasonable specification of the position?

- `30`km away
- `30`km away at `4`km altitude above sea level
- `30`km to the east, `40`km north, and at `4`km altitude above sea level
- `30`km to the east, `40`km north, and at `4`km altitude above sea level

Answer is to specify - North, East, and Altitude. With these three pieces of information, the position of the airplane is specified clearly.

We have arrived at representing vectors that have magnitude and direction.

The representation is

`30`km East + `40`km North + `4`km Altitude

This can be generalized to a 3D space with three axes x-axis, y-axis, and z-axis. The figure represents x-axis, y-axis, and z-axis. A point 'p' is shown.

Do you understand the figure? What is the x, y, and z coordinates, given in `(x,y,z)` order, of the point given in figure?

- `(7,4,10)`
- `(4,7,10)`
- `(4,7,10)`
- `(10,7,4)`
- `(4,10,7)`

The answer is '`(4,7,10)`'.

The point is projected onto x-y plane. That is further projected onto x and y axes.

The airplane can be considered as the point 'p' in the three dimensional space. The representation is simplified as

`4i+7j+10k`

or alternatively

`4 hat i+7 hat j+10 hat k`

Where

`i` or `hat i` represents the direction in `x`-axis

`j` or `hat j` represents the direction in `y`-axis

`k` or `hat k` represents the direction in `z`-axis

Note: `hat i` is pronounced as i-hat or i-cap.

A vector quantity is represented with **3 components** along the three directions of 3D coordinates.

**Mathematical representation: ** A vector quantity is represented in the form `ai+bj+ck`

or alternatively

`a hat i+b hat j+c hat k`

Where i, j, k are the directions along x, y, and z axes

and a, b, c are the magnitude along the directions respectively.

*Solved Exercise Problem: *

Represent OP as a vector.

- `3i+5j`
- `3i+5j`
- `8`
- `5i+3j`
- `sqrt(3^2+5^2)`

answer is '`3i+5j`'.

The x-axis component is represented with an `i` and y-axis component is represented with a `j`.

*Solved Exercise Problem: *

What is the vector form of `bar(OP)`?

- `6i+3j+4k`
- `3i+6j+4k`
- `3i+6j+4k`
- `4i+3j+6k`
- `4i+6j+3k`

answer is '`3i+6j+4k`'. Referring to the figure, the component along each of the axes

`3i`: `3` along x axis

`6j`: `6` along y axis

`4k`: `4` along z axis

*Solved Exercise Problem: *

What is the vector form of OP?

- `-11/2 i-1.3j+4k`
- `4i-11/2 j-1.3k`
- `-1.3i-11/2 j-4k`
- `-1.3i-11/2 j+4k`
- `-1.3i-11/2 j+4k`

answer is '`-1.3i-11/2 j+4k`'.

*Solved Exercise Problem: *

What is the vector form of OP?

- `2.4i+ 7/2 j`
- `2.4i+ 7/2 j+0k`
- both the above
- both the above
- none of the above

answer is 'both the above'. When a two dimensional vector is presented, the component along the third dimension is 0.

*Solved Exercise Problem: *

A point is located from the x, y, z-axes at distances `2`, `-1.2`, and `1.4` units respectively. What is the vector representation of the point?

- `2−1.2+1.4`
- `2.2`
- `sqrt(2^2+1.2^2+1.4^2)`
- `2i-1.2j+1.4k`
- `2i-1.2j+1.4k`

answer is '`2i-1.2j+1.4k`'

*Solved Exercise Problem: *

What does a `vec(OP) = −.5i+2.1j−.6k` mean?

- the point P is `−0.5` unit away from x-axis
- the point P is `2.1` unit away from y-axis
- the point P is `−0.6` unit away from z-axis
- all the above
- all the above

answer is 'all the above'

*slide-show version coming soon*