Vectors and Scalars: Concepts and Operations in Physics

Vectors and Scalars

Introduction to Scalars and Vectors

Physical quantities in physics are classified as either scalars or vectors. Understanding the distinction between these two types is fundamental for describing motion, forces, and other phenomena.

• Scalar Quantities: Quantities described completely by a single number (magnitude) and a unit. Examples include length, time, temperature, mass, and speed.

• Vector Quantities: Quantities described by both a magnitude and a direction. Examples include displacement, velocity, acceleration, and force.

• Notation: Scalars are written using non-bold italic letters (e.g., m, t, d), while vectors are written using bold, non-italic letters (e.g., x, F) or letters with an arrow above them (e.g., $\vec{x}$, $\vec{F}$).

Example: Temperature is a scalar (e.g., 25°C), while velocity is a vector (e.g., 5 m/s east).

Graphical Addition of Vectors

Representing Vectors

Vectors are represented in diagrams by arrows. The direction of the arrow shows the direction of the vector, and the length of the arrow is proportional to the magnitude.

• To specify a vector, state its magnitude and the angle it makes with a reference direction (e.g., positive x-axis or compass direction).

• Example: A car moving at 5.0 m/s at an angle of 37° south of west.

Tip-to-Tail Method

To add two or more vectors graphically, use the tip-to-tail method:

1. Draw the first vector to scale in the correct direction.

2. Draw the second vector so that its tail starts at the tip of the first vector.

3. The resultant vector (sum) is drawn from the tail of the first vector to the tip of the last vector.

• Commutativity: Vector addition is commutative: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$.

• Resultant: The sum of two vectors is called the resultant.

• Important: Never add only the magnitudes; directions must be considered.

Parallelogram Method

An alternative graphical method for adding two vectors:

1. Draw both vectors from the same starting point.

2. Construct a parallelogram using the vectors as adjacent sides.

3. The resultant vector is drawn from the starting point to the opposite corner of the parallelogram.

• Note: The parallelogram method is only used for adding two vectors.

Subtracting Vectors

Vector Subtraction

To subtract vector $\vec{A}$ from $\vec{B}$, add the negative of $\vec{A}$ to $\vec{B}$:

• The negative of a vector $-\vec{A}$ has the same magnitude as $\vec{A}$ but points in the opposite direction.

• Formula: $\vec{B} - \vec{A} = \vec{B} + (-\vec{A})$

• Graphically, draw $-\vec{A}$ and use tip-to-tail or parallelogram methods as with addition.

Multiplying a Vector by a Scalar

Scalar Multiplication

Multiplying a vector $\vec{A}$ by a scalar $c$:

• If $c$ is positive, $c\vec{A}$ points in the same direction as $\vec{A}$ and has magnitude $|c|$ times that of $\vec{A}$.

• If $c$ is negative, $c\vec{A}$ points in the opposite direction and has magnitude $|c|$ times that of $\vec{A}$.

• Example: $2\vec{A}$ is twice as long as $\vec{A}$ in the same direction; $-2\vec{A}$ is twice as long but in the opposite direction.

Adding Vectors by Components

Component Method

For quantitative accuracy, vectors are often added or subtracted by their components. Any vector in the x-y plane can be expressed as the sum of its x and y components:

• Vector Components: $\vec{A} = A_x + A_y$

• $A_x$ is the component along the x-axis, $A_y$ along the y-axis.

• Tip-to-tail addition of $A_x$ and $A_y$ gives the original vector $\vec{A}$.

Finding Components Using Trigonometry

Given a vector $\vec{A}$ with magnitude $A$ and angle $\theta$ with respect to the positive x-axis:

• $A_x = A \cos \theta$

• $A_y = A \sin \theta$

• Trigonometric relationships:

• $\sin \theta = \frac{\text{opp}}{\text{hyp}}$

• $\cos \theta = \frac{\text{adj}}{\text{hyp}}$

• $\tan \theta = \frac{\text{opp}}{\text{adj}}$

• $h^2 = a^2 + o^2$ (Pythagorean theorem)

Adding and Subtracting Vectors by Components

Given two vectors $\vec{A}$ and $\vec{B}$:

• Sum: $\vec{C} = \vec{A} + \vec{B}$ has components:

  • $C_x = A_x + B_x$

  • $C_y = A_y + B_y$

• Difference: $\vec{C} = \vec{A} - \vec{B}$ has components:

  • $C_x = A_x - B_x$

  • $C_y = A_y - B_y$

• Multiplication by a Scalar: $\vec{C} = c\vec{A}$ has components:

  • $C_x = cA_x$

  • $C_y = cA_y$

Equation Summary

Operation	Equation (LaTeX)	Description
Addition (Tip-to-Tail)	$\vec{C} = \vec{A} + \vec{B}$	Resultant vector from tip-to-tail addition
Component Addition	$C_x = A_x + B_x$ $C_y = A_y + B_y$	Add x and y components separately
Component Subtraction	$C_x = A_x - B_x$ $C_y = A_y - B_y$	Subtract x and y components separately
Scalar Multiplication	$C_x = cA_x$ $C_y = cA_y$	Multiply each component by scalar $c$
Finding Components	$A_x = A \cos \theta$ $A_y = A \sin \theta$	Use trigonometry to resolve vector into components

Summary

• Scalars have magnitude only; vectors have both magnitude and direction.

• Vectors can be added graphically (tip-to-tail, parallelogram) or by components.

• Subtracting a vector is equivalent to adding its negative.

• Multiplying a vector by a scalar changes its magnitude and possibly its direction.

• Trigonometry is used to resolve vectors into components for calculation.