In this page, the primary application of vector dot product "cause-effect pairs" is explained.

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In this chapter, the product defined for components in parallel is discussed. First, let us establish the application scenario where components in parallel interact to form a product.

A pen can be used to write `30` pages. How many pages one can write with `4` pens?

- `4xx30`
- `120`
- both the above
- both the above

Answer is 'both the above'. In this 'number of pen' is a *cause* and 'writing a number of pages' is an *effect*.

This is an example of *cause and effect pair* in scalar quantities.

Can you identify a cause-effect pair in the following?

- Volume of Paint and painted area
- Number of tickets sold and the money collected in the sale
- speed of a car and distance covered in an hour
- all the above
- all the above

The answer is 'all the above'.

Similar to the cause-effect pairs of scalar quantities, vectors have cause and effect pairs.

Can you identify the cause-effect pair in vector quantities?

- pull a table and the table moves
- pull a table and the table moves
- electronic item heats up when they are used
- click a photo using a mobile
- none of the above

Answer is 'pull a table and the table moves'. Pulling force has direction and the movement of the table has direction. So, Both of them are vector quantities.

Can you identify a cause-effect pair in vector quantities?

- velocity causes displacement
- acceleration causes velocity
- force causes acceleration
- all the above
- all the above

The answer is 'all the above'.

Two vector quantities may be in relation as ** cause-effect pair**.

One pulls a table towards east direction, to which direction will the table move? (Given that no other force is acting on the table.)

- towards east
- towards east
- towards west
- towards north
- towards north-east

Answer is 'towards east'.

One pulls a table in a direction. Is it possible that the table move in any direction other than the direction of force? (given that no other force is acting on the table.)

- Not Possible
- Not Possible
- it is Possible

Answer is 'Not possible'. *If the force is in one direction, then the displacement due to that force is in the same direction.*

A ball moves with velocity `vec v`. In which direction will the displacement due to that velocity be?

- in the same direction as `vec v`
- in the same direction as `vec v`
- in perpendicular to `vec v`
- in any direction
- none of the above

Answer is 'in the same direction as `vec v`'.

A ball moves with velocity `vec v_1` in one direction and velocity `vec v_2` in another direction. In which direction will the displacement `vec s_1` due to `vec v_1` be?

- in the direction of `vec v_1`
- in the direction of `vec v_1`
- in the direction of `vec v_2`
- in the direction of `vec v_1 + vec v_2`
- none of the above

Answer is 'in the direction of `vec v_1`'.

A ball moves with velocity `vec v_1` in one direction and velocity `vec v_2` in another direction. In which direction will the displacement `vec s_2` due to `vec v_2` will be?

- in the direction of `vec v_1`
- in the direction of `vec v_2`
- in the direction of `vec v_2`
- in the direction of `vec v_1 + vec v_2`
- none of the above

Answer is 'in the direction of `vec v_2`'.

A ball moves with velocity `vec v_1` in one direction and velocity `vec v_2` in another direction. The sum of velocities `vec v = vec v_1 + vec v_2`. In which direction will the displacement `vec s` due to `vec v` be?

- in the direction of `vec v_1`
- in the direction of `vec v_2`
- in the direction of `vec v_1 + vec v_2`
- in the direction of `vec v_1 + vec v_2`
- none of the above

Answer is 'in the direction of `vec v_1 + vec v_2`'.

Note that the displacement component is in the same direction as the velocity component causing it.

• `vec s_1` is in the direction of `vec v_1`

• `vec s_2` is in the direction of `vec v_2`

• `vec s_1+vec s_2` is in the direction of `vec v_1+vec v_2`

*The result, due to a cause, will only be in the direction of the cause. *

**Direction Property of Cause-Effect pairs :** When a vector quantity `vec x` results in another vector quantity `vec y` , the direction of the result `vec y` will be same as that of the cause `vec x`.

One pulls a table towards east and at the same time, another pulls the same table towards north. If both are pulling with same magnitude, what direction will the table move?

- North
- East
- North-East
- North-East
- South-West

Answer is 'North-East'

This can be verified with a simple experiment at home / class.

The problem is shown in the figure. The forces `vec f_1` and `vec f_2`. The combined `vec f_r` is shown. `vec f_r = vec f_1+vec f_2`. Which law of vector addition is used in this?

- Parallelogram law of Vector Addition
- Triangular law of Vector Addition
- Both the above can be used interchangeably
- Both the above can be used interchangeably

Answer is 'Both the above can be used interchangeably'. Triangular law or Parallelogram law are to understand different aspects of vector addition, and both equivalently describe vector addition.

The displacement `vec s`, due to force `vec f`, is shown in the figure. Which of the following is understood from the figure?

- `vec f_1` causes `vec s_1` in the same direction
- `vec f_2` causes `vec s_2` in the same direction
- `vec f_r` causes `vec s_r` in the same direction
- all the above
- all the above

Answer is 'All the above'.

If force `vec f` causes `vec s` displacement, then the work done by the force is calculated as

work = `f` multiplied by `s`

Given the force, displacement pairs in the figure, What is the work done by force `vec f_r`?

- `f_r` multiplied by `s_r`
- `f_r` multiplied by `s_r`
- `f_r` multiplied by `s_1`
- `f_r` multiplied by `s_2`

Answer is '`f_r` multiplied by `s_r`'. Because `f_r` causes `s_r`.

Given the force, displacement pairs in the figure, What is the work done by force `vec f_1`?

- `f_1` multiplied by `s_r`
- `f_1` multiplied by `s_r`
- `f_1` multiplied by `s_1`
- `f_1` multiplied by `s_2`

Answer is '`f_1` multiplied by `s_1`'. Because `f_1` causes `s_1`.

The force and displacement pair are given as in the figure. Note that only force `vec f_1` and displacement `s_r` are given. It is understood that some unknown force is also acting on the object. That is the reason the displacement is not in the same direction as the force. What is the work done by force `vec f_1`?

- magnitudes of `f_1` and `s_r` multiplied
- cannot compute using only the magnitudes
- cannot compute using only the magnitudes

Answer is 'Cannot compute using only the magnitudes'

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