# Scalar Quantities, Vector Quantities & Finding resultant of two perpendicular forces

Physical quantities can be classified into Scalar and Vector Quantities , Scalar quantity is a physical quantity that can be fully defined by its magnitude only & it has no direction while Vector quantity is a physical quantity that can be fully defined by both magnitude and direction .

Examples of Scalar quantity : Distance , Mass , Time , Temperature & Energy  , Examples of Vector quantity : Displacement , Velocity , Acceleration & Force .

When measuring a physical quantity such as Temperature ( The magnitude of temperature say , 27 ° C describes temperature fully ) , Velocity ( A value of 20 km/h does not describe fully the velocity of a car such that the direction of the car motion should be defined ) .

### Distance and Displacement

There is a difference between the concept of displacement and the concept of distance , Distance is the length of the path moved by an object from a position to another , Distance is a scalar quantity that can be fully defined by its magnitude only .

Displacement is the length of the straight line segment in a given direction between the starting point and the end point , Displacement is a vector quantity that can be fully defined by its magnitude and direction .

#### Guidelines to solve problems

• If an object moves in one direction from A to B , the magnitude of displacement equals the distance covered .
• If an object moves in one direction from A to B then returns back to A , the magnitude of displacement = 0 and the distance covered = 2 AB .
• If an object moves in a curved path from A to B , Displacement would be shorter than distance .
• If an object moves in one direction from A to B and then reverses its direction to C , then the displacement ( d ) = AB – BC , The distance ( s ) = AB + BC .
##### Representation of vector quantities

The vector quantity is represented by a directed straight segment () whose base is at the starting point and its tip is at the end point , where its length is proportional to the vector magnitude , The arrow direction points to the direction of the vector quantity , The vector quantity is denoted by a bold letter ( A ) or a letter tagged by a small arrow .

##### Some basics & vector algebra

Two vectors are equal when they have the same magnitude and the same direction ( even if they have different starting points ) , Two vectors are not equal when they have different directions ( even if they have the same magnitude ) or different magnitudes ( even if they the same direction ) .

##### Vector Algebra

Vector Algebra such as Vectors addition , Vector resolution , Vector product ( Scalar ( dot ) product and Cross product ) .

##### Resultant ( addition ) of vectors

When two forces or more act on an object , this object would move in a certain direction determined by the resultant of these forces , The resultant force is a single force that produces the same effect on an object as that produced by the original acting forces .

##### Application

If a rock is pulled by two ropes with two forces of 30 N and 40 N having an angle 90° between them , We notice that the rock is moved a certain distance in a direction different than those of the two forces ( during a certain time ) .

If the two ropes are replaced by one rope and pulled by a force of 50 N , We notice that the rock is moved through the same distance in the same direction when it is affected by the two forces during the same time .

This means that the force 50 N produces the same effect as that of the two forces 30 N and 40 N , So , it is considered the resultant of the two forces 30 N and 40 N .

There are two ways to add two vectors by drawing a triangle and by drawing a parallelogram in which A and B are adjacent sides , Thus , the diagonal represents their resultant .

##### First : Graphically
1. Draw a horizontal line ( AB ) , on the graph paper , of length 3 cm to represent the first force ( F1 = 3 N ) .
2. Perpendicular to ( A B ) at the point ( A ) , draw a vertical line ( AD ) of length 4 cm to represent the second force ( F2 = 4 N ) .
3. Complete the rectangle ABCD .
4. Join the diagonal ( AC ) to represent the magnitude and direction of the resultant .
5. Measure the length of the line segment ( AC ) that represents the magnitude of the resultant force .
6. Measure the angle ( BAC ) that defines the direction of the resultant force relative to the first force ( F1 ) .
##### Second : Theoretically

Find the magnitude of the resultant force using Pythagoras’ theorem for the right angled triangle ( AC² = AB² + BC² ) .

F² = F1² + F2²

We can find the angle ( θ ) by the relation

tan θ = Fy / Fx = F2 / F1

##### Resolution of a vector

Resolution of a vector is the reverse operation for getting the resultant of some vectors where a force can be resolved into two perpendicular forces along dimensions ( x , y ) , Thus :

Fy = F sin θ

Fx = F cos θ

##### Product of vectors

There are different forms of finding the product of two vectors :

##### Scalar ( dot ) product

The dot product of two vectors A and B is expressed as follows :

A . B = A B cos θ

The sign ( . ) is pronounced ” dot ” , The result is a scalar quantity .

The work is a scalar quantity as it is the dot product of two vectors which are force and displacement , W = F . d , If the vector ( F ) acts on the vector ( d ) at angle ( θ ) .

W = F D cos θ

##### Vector ( cross ) product

The cross product of two vectors A and B is expressed as follows :

C = A ^ B = A B sin θ n

Where n is a unit vector perpendicular to the plane of both vectors A and B , The sign ( ^ ) is pronounced cross and the result is a vector quantity C , The vector C points to the direction of n perpendicular to the plane of both vectors A and B .

The operation of the electric motor depends on the presence of two vectors which are electric field and magnetic field , while they cause the rotation of the motor coil in a direction perpendicular to the plane of them both .

##### The right hand rule

It is used to define the direction of the vector product C of two vectors A and B .

##### How to apply ?

Move the fingers of the right hand from the first vector towards the second vector through the smaller angle between them θ , the thumb then points to the direction of their vector product .

Static objects , Moving objects , Types of Motion and Velocity