How the property 'direction' affects vector quantities in mathematical calculations?

*click on the content to continue..*

The fundamental mathematical operations for vectors are introduced. Let us see how having direction component affects these fundamental operations in vectors.

A person walks `5`m east and then takes the following path

• `3`m north

• `4.2`m south

• `3/4` m north

At this position, how far is the person away from the starting point *in the east direction* ?

- `5`m
- `5`m
- `5+3+4.2+3/4`m
- `5+3-4.2+3/4`m
- none of the above

Answer is '`5`m '– as the person moved `5` meter east and then all his movements were in directions north and south.

Note that *any change in north and south direction does not affect the magnitude in east direction.*

*Any change in a direction affects component along that direction only* and does not affect the components in the directions at `90^@` to that direction.

**Independence of Quantities along orthogonal directions: ** For a vector, changes along one axis affect only the component along that axis and do not affect the components along other axes, as the axes are orthogonal.

A ball has a velocity in x-axis `20` m/sec. It also has an unknown velocity in y-axis. In `2` seconds, how far the ball would travel along x-axis?

- cannot calculate as the velocity along y-axis is not given
- `20xx2` along x-axis
- `20xx2` along x-axis

Answer is '`20xx2`' along the x-axis. The velocity causes displacement. The result 'displacement' in one direction is only caused by the component of 'velocity' in that direction.

A person walks `5`m at `30^@` angle to east. Then he turns `90^@` clockwise and walks for 2m. What is the distance from the starting point along the direction `30^@` angle to east?

- `5`m
- `5`m
- `7`m
- `sqrt(5^2+7^2)`
- none of the above

Answer is '`5`m' – as the person moved only `5` meter in the direction `30^@` angle to east. He walked in perpendicular to the direction after that.

A person walks `5`m at `30^@` angle to east. Then he turns `40^@` clockwise and walks for 2m. What is the distance from the starting point along the direction `30^@` angle to east?

- `5`m
- `7`m
- `sqrt(5^2+7^2)`
- none of the above
- none of the above

Answer is 'none of the above' – The `2`m walk is not at right angle to the direction in question.

This is explained with an illustration in the next page.

Take two persons `A` and `B` start from the point `o` and walks to `p` in direction the `d`.

• Person `A` turns at right angle and walks to `q`

• Person B turns at an angle `40^@` and walks to `r`

The displacement along direction `d` is asked.

• For person `A`, the displacement along `d` is between points `o` and `p`

• For person `B` the displacement along `d` is between points `o` and `s`, as the segment `bar(pr)` has component `bar(ps)` along direction `d`.

The component which is in perpendicular to a direction does not affect the component in the direction. The directions which are at `90^@` angle are referred to as orthogonal directions.

In arithmetic operations like addition, subtraction, multiplication by a scalar, and product of vectors, – the results are computed for

• **components in parallel**

• **components in perpendicular **

Each of these two components interact differently.

How does direction affect vector quantities in multiplication? At an abstract level, there are two products possible. Given multiplicand `vec p` and multiplier `vec q`. `vec q` is split into `vec a` and `vec b`, such that

`vec q = vec a + vec b` and

`vec a` is in parallel to `vec p`

`vec b` is in perpendicular to `vec p`

two forms of multiplications are defined for each of these two components.

• one with component in parallel to the other, called vector dot product.`vec p cdot vec q = vec p cdot vec a`

• another with component in perpendicular to the vector, called vector cross product. `vec p times vec q = vec p times vec b`

**Effect of direction: ** In mathematical calculations, vectors have the following properties

• A vector is represented as components along orthogonal directions.

• In vector addition, components in parallel add up and components in perpendicular are combined using Pythagoras Theorem.

• Two types of vector multiplication are defined,

• In vector dot product, only the component of the vector in parallel to the other vector takes part in multiplication.

• In vector cross product, only the component of the vector in perpendicular to the other vector takes part in multiplication.

*Solved Exercise Problem: *

What is repeated addition of a vector?

- Multiplication of vector by Scalar
- Multiplication of vector by Scalar
- Multiplication of magnitude of vector without direction

The answer is 'Multiplication of vector by Scalar'

*slide-show version coming soon*