# Scalar Quantities , Vector Quantities and Finding the 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 .**

**Finding the resultant of two perpendicular forces **

**First : Graphically **

**Draw a horizontal line ( AB ) , on the graph paper , of length 3 cm to represent the first force ( F1 = 3 N ) .****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 ) .****Complete the rectangle ABCD .****Join the diagonal ( AC ) to represent the magnitude and direction of the resultant .****Measure the length of the line segment ( AC ) that represents the magnitude of the resultant force .****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 = F**_{1}² + F_{2}²

_{1}² + F

_{2}²

**We can find the angle ( θ ) by the relation**

** tan θ = F**_{y} / F_{x} = **F**_{2 }**/ ****F**_{1}

_{y}/ F

_{x}=

_{2 }

_{1}

**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 :**

**F**_{y} = F sin θ

_{y}= F sin θ

**F**_{x} = F cos θ

_{x}= 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**

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