Nature of quantities

2.3.1 Scalars and vectors:

2.3.1 (a) Scalar and vector quantities:

• Scalars and vectors are two types of quantities used to describe physical properties and mathematical objects.

⇒ Scalars quantity:

• Are quantities with only magnitude (size or amount)
• Have no direction
• Can be described by a single number

Table 1 Scalar quantities

⇒Vectors Quantity:

• Are quantities with both magnitude and direction.
• Can be described by a magnitude (length) and a direction (arrow)
• Scalars are described by a single number, while vectors are described by a magnitude and direction.
• Scalars are invariant under coordinate transformations, while vectors transform according to specific rules.

Table 2 Vector quantities

Examples
(1)
weight is a force and has a size that we can calculate from the equation weight = mass × gravitational field strength.

[math]W = m × g [/math]

On Earth, the value for g is about [math] 9.8 Nkg^{-1}[/math],
So, the weight of a person of mass 78 kg is about 7645 N, downwards. Weight always acts towards the center of the Earth, so it has a direction associated with it.
(2)
Speed is a scalar quantity; velocity is a vector quantity.
Imagine a 400 m race in athletics. The average speed of the athlete is given by the equation average

[math] Speed = \frac{distance}{time}[/math]

While the velocity,

[math]Velocity = \frac{displacement}{time}[/math]

So, if the athlete runs 400 m in 40 seconds, then the average speed is 10[math]m.s^{-1}[/math].

2.3.1 (b) Vector addition and subtraction:

⇒Vector addition:

• Vector addition is a way of combining two or more vectors to get a resultant vector.
• The resultant vector is the vector that connects the tail of the first vector to the head of the second vector.
• Vector addition is commutative, meaning that the order of the vectors does not change the result.
• Vector addition is associative, meaning that you can add multiple vectors in any order.
• Addition of two vectors are [math]\vec{A}, \vec{B}, \text{and} \vec{R}[/math], represented by resultant vector.
• We calculate the result vector
• [math]\vec{R} = \vec{A} + \vec{B}[/math]
• 
• Figure 1 Addition of two vectors
• There are two methods to add vectors:
• Graphical Method: This method involves drawing the vectors tail-to-tail and connecting the tail of the second vector to the head of the first vector. The resultant vector is the vector from the tail of the first vector to the head of the second vector.
• 

  Figure 2 Vector addition by tail-to-tail

• Component Method: This method involves breaking down each vector into its x and y components, adding the x components and y components separately, and then combining the resulting x and y components to get the resultant vector.
• 

  Figure 3 Vector addition by component method

• Vector subtraction:
• Vector subtraction is a mathematical operation that combines two vectors by subtracting the corresponding components. It’s a way to find the difference between two vectors.
• 

  Figure 4 Vector subtraction

• Given two vectors:
• [math]\vec{A}= (a1, a2, …, an) \\ \vec{B} = (b1, b2, …, bn)[/math]
• The subtraction of from [math]\vec{B}[/math] from [math]\vec{A}[/math] is:
• [math]\vec{A} – \vec{B}[/math]= (a1 – b1, a2 – b2, …, an – bn) [/math]
• This results in a new vector, which represents the difference between the original vectors.
• Properties of vector subtraction:
• – Commutativity:[math]\vec{A} – \vec{B} \neq \vec{B} – \vec{A}[/math] (except when [math]\vec{B}[/math] is the zero vector)
• – Associativity:[math] (\vec{A} – \vec{B}) – \vec{C} = \vec{A} – (\vec{B} – \vec{C}) [/math]
• – Distributivity: [math] \vec{A} – (\vec{B} + \vec{C}) = \vec{A} – \vec{B} – \vec{C}[/math]
• Figure 5 Vector subtraction by graphically representation of two vectors ([math]\vec{A}, \vec{B}[/math]

2.3.1 (c) Vector triangle:

• Obviously, vectors do not always act parallel or anti-parallel to one another. Figure 6 shows an aircraft cruising at a speed of [math]360 ms^{-1}[/math] north with the wind blowing at a speed of [math]85 ms^{-1}[/math] from the east.
• Figure 6 an aircraft cruising its speed
• We can work out the ‘vector sum’ of these velocities to find the actual speed and direction of the aircraft.
• Since the two vectors act as right angles to each other we use Pythagoras’ theorem and trigonometry to calculate the magnitude of the resultant vector [math]V_R[/math].
• [math]V_R = \sqrt{V_\text{wind}^2 + V_\text{airspeed}^2} \\
  V_R = \sqrt{(7225)^2 + (129600)^2} \\
  V_R = 370 \, \text{ms}^{-1}[/math]
• use trigonometry to determine the direction of [math]V_R[/math]:
• [math]\tan \theta = \frac{V_\text{wind}}{V_\text{airspeed}} \\
  \theta = \tan^{-1} \left( \frac{V_\text{wind}}{V_\text{airspeed}} \right) \\
  \theta = \tan^{-1} \left( \frac{85}{360} \right) \\
  \theta = 13.3^\circ [/math]

⇒Adding vectors that are not at right angles:

• Vector triangles are also used to find the resultant of two vectors not at right angles.
• However, Pythagoras’ theorem cannot be used in these cases.
• We must make a scale drawing of the vector triangle and then use a ruler to measure the size of the resultant and a protractor to find its direction.
• Always state the angle with reference to a specified direction, for example measured clockwise from north or anticlockwise from the positive x direction.

Examples
(1)
Calculate the resultant velocity and the angle shown in Figure 7

Figure 7 Find resultant vector[math]V_R[/math]

Given data:
X-component [math]V_X = 30 ms^{-1}[/math]
Y-component [math]V_Y = 0 ms^{-1}[/math]

Find data:
Resultant vector  [math]V_R [/math]=?
Direction angle θ =?
Formula:

1. [math]V_R = \sqrt{V_X^2 + V_Y^2} [/math]
2. [math] \theta = \tan^{-1} \left( \frac{V_X}{V_Y} \right) [/math]

Solution:

1. For resultant vector:

   [math]V_R = \sqrt{V_X^2 + V_Y^2} \\
   V_R = \sqrt{(30)^2 + (40)^2} \\
   V_R = \sqrt{2500} \\
   V_R = 50 \, \text{ms}^{-1}[/math]

2. For direction angle:

   [math]\theta = \tan^{-1} \left( \frac{V_X}{V_Y} \right) \\
   \theta = \tan^{-1} \left( \frac{30}{40} \right) \\
   \theta = \tan^{-1} (0.75) \\
   \theta = 36.87^\circ [/math]

2.3.1 (d) Resolving vector:

• Resolving a vector means breaking it down into its component parts, typically into two perpendicular components. This is also known as “decomposing” a vector.
• There are two common ways to resolve a vector:
• Horizontal and Vertical Components (also known as x and y components):
• – Resolve a vector into its horizontal (x) and vertical (y) components. This is useful for vectors on a 2D plane.
• Rectangular Components (also known as i and j components):
• Resolve a vector into its rectangular components, typically represented by the unit vectors i and j. This is useful for vectors in a 2D coordinate system.
• Mathematically:
• [math]\vec{A} = (A_x, A_y) = A_x \, \hat{i} + A_y \, \hat{j} [/math]
• Where:
• – [math]\vec{A_x}[/math]is the horizontal (x) component
• – [math]\vec{A_y}[/math]is the vertical (y) component
• – i is the unit vector in the x-direction
• – j is the unit vector in the y-direction

  

• Resolving a vector involves the production of two vectors, at right angles to one another, the sum of which is equal to the original vector. These vectors are often shown as the horizontal component and the vertical component of the original vector.
• When you resolve a resultant vector into its horizontal and vertical components.
• The reason behind this is that the vertical component determines the vertical behavior of the object’s motion only, and the horizontal component describes the object’s horizontal motion only.
• In other words, the vertical and horizontal components of an object’s motion are independent – they have no effect on each other at all.
• This is because the angle between the two components is 90° and cos 90° = 0, so the value of the horizontal component is zero when we consider the vertical component, and vice versa.

Figure 8 Resolution of forces

• The components OB and BA are perpendicular to each other. They are called the perpendicular components of OA representing force F. Hence OB represents its x-component [math] F_x [/math] and BA represents its x y-component [math]F_y [/math].
• [math] F = F_x + F_y \\
  F_x = F \cos \theta \\
  F_y = F \sin \theta [/math]