In Lesson 2, we studied motion along a straight line. Such motion is called one-dimensional motion because only one coordinate is needed to specify the position of an object.

However, most motion in nature occurs in two or three dimensions.

Consider the following examples:

• A soccer ball kicked through the air.
• A drone flying above a city.
• Blood flowing through branching arteries.
• A warehouse robot navigating around shelves.
• Earth orbiting the Sun.

To describe such motions mathematically, we need a new concept called a vector.

Vectors are among the most important mathematical objects in physics. They appear in:

• Classical mechanics
• Electromagnetism
• Fluid mechanics
• General relativity
• Quantum mechanics
• String theory

In fact, when we eventually study quantum mechanics, quantum states themselves will be represented by generalized vectors in an abstract mathematical space called a Hilbert space.

This lesson introduces vectors and shows how they allow us to describe motion in two dimensions.

---

Scalars and Vectors

Physical quantities can be classified into two broad categories.

Scalars

A scalar possesses magnitude only.

Examples include:

• Mass
• Temperature
• Time
• Energy
• Distance
• Speed

Suppose a car travels at

$$ 60 \text{ km/h} $$

This tells us how fast the car is moving but does not specify the direction of motion.

Therefore speed is a scalar quantity.

---

Vectors

A vector possesses both magnitude and direction.

Examples include:

• Velocity
• Acceleration
• Force
• Momentum
• Electric field

Suppose a car travels at

$$ 60 \text{ km/h East} $$

Now both the magnitude and direction are known.

Therefore velocity is a vector quantity.

---

Geometric Representation of Vectors

Vectors are represented graphically by arrows.

The length of the arrow represents magnitude.

The direction of the arrow represents direction.

For example, if a vector points northeast and has a magnitude of 5 units, it can be represented by an arrow of length 5 pointing northeast.

This geometric interpretation forms the basis for vector operations.

---

Components of a Vector

Suppose a vector points diagonally upward.

Instead of treating it as a single object, we can decompose it into:

• Horizontal component
• Vertical component

Suppose

$$ \vec{A}=(3,4) $$

This means:

• 3 units in the horizontal direction
• 4 units in the vertical direction

The pair of numbers uniquely specifies the vector.

---

Magnitude of a Vector

The magnitude of a vector is found using the Pythagorean theorem.

For a vector

$$ \vec{A}=(A_x,A_y) $$

its magnitude is

$$ |\vec{A}|=\sqrt{A_x^2+A_y^2} $$

---

Example

Consider

$$ \vec{A}=(3,4) $$

Then

$$ |\vec{A}|=\sqrt{3^2+4^2} $$

$$ |\vec{A}|=\sqrt{9+16} $$

$$ |\vec{A}|=5 $$

Thus the vector has magnitude 5.

---

Direction of a Vector

The direction of a vector can be obtained from trigonometry.

Suppose the vector components are

$$ A_x $$

and

$$ A_y $$

Then

$$ \tan(\theta)=\frac{A_y}{A_x} $$

Therefore

$$ \theta=\tan^{-1}\left(\frac{A_y}{A_x}\right) $$

---

Example

For the vector

$$ \vec{A}=(3,4) $$

the direction angle is

$$ \theta=\tan^{-1}\left(\frac{4}{3}\right) $$

$$ \theta=53.13^\circ $$

Thus the vector points approximately 53 degrees above the positive x-axis.

---

Unit Vectors

A unit vector is a vector whose magnitude equals one.

The standard unit vectors are

$$ \hat{i} $$

for the x-direction and

$$ \hat{j} $$

for the y-direction.

Their magnitudes satisfy

$$ |\hat{i}|=1 $$

and

$$ |\hat{j}|=1 $$

Unit vectors allow us to express vectors in a compact algebraic form.

---

Vector Representation Using Unit Vectors

The vector

$$ \vec{A}=(3,4) $$

may be written as

$$ \vec{A}=3\hat{i}+4\hat{j} $$

This notation is used extensively throughout physics and engineering.

---

Vector Addition

Suppose

$$ \vec{A}=(3,4) $$

and

$$ \vec{B}=(2,1) $$

Then vector addition is performed component-wise:

$$ \vec{A}+\vec{B}=(3+2,4+1) $$

Thus

$$ \vec{A}+\vec{B}=(5,5) $$

---

Vector Subtraction

Suppose

$$ \vec{A}=(6,4) $$

and

$$ \vec{B}=(2,1) $$

Then

$$ \vec{A}-\vec{B}=(6-2,4-1) $$

Thus

$$ \vec{A}-\vec{B}=(4,3) $$

---

Resolving a Vector into Components

Suppose a vector has magnitude

$$ A $$

and direction

$$ \theta $$

Its horizontal component is

$$ A_x=A\cos\theta $$

Its vertical component is

$$ A_y=A\sin\theta $$

---

Example

Suppose

$$ A=10 $$

and

$$ \theta=30^\circ $$

Then

$$ A_x=10\cos(30^\circ) $$

$$ A_x=8.66 $$

and

$$ A_y=10\sin(30^\circ) $$

$$ A_y=5 $$

---

Position Vectors

The position of an object in two dimensions is represented by the position vector

$$ \vec{r}=x\hat{i}+y\hat{j} $$

where

$$ x $$

is the horizontal position and

$$ y $$

is the vertical position.

Position vectors allow us to describe the location of an object mathematically.

---

Velocity in Two Dimensions

Velocity is also a vector quantity.

The velocity vector is

$$ \vec{v}=v_x\hat{i}+v_y\hat{j} $$

The speed is the magnitude of the velocity vector:

$$ |\vec{v}|=\sqrt{v_x^2+v_y^2} $$

---

Acceleration in Two Dimensions

Acceleration is similarly written as

$$ \vec{a}=a_x\hat{i}+a_y\hat{j} $$

with magnitude

$$ |\vec{a}|=\sqrt{a_x^2+a_y^2} $$

---

Projectile Motion

Projectile motion is one of the most important applications of two-dimensional motion.

Examples include:

• Kicked soccer balls
• Thrown baseballs
• Arrows
• Water fountains
• Artillery shells

A projectile experiences two independent motions:

1. Horizontal motion
2. Vertical motion

---

Fundamental Assumptions

Ignoring air resistance:

Horizontal acceleration is

$$ a_x=0 $$

Vertical acceleration is

$$ a_y=-g $$

where

$$ g=9.81\text{ m/s}^2 $$

---

Launch Velocity Components

Suppose a projectile is launched with initial speed

$$ v_0 $$

at angle

$$ \theta $$

Then

$$ v_{0x}=v_0\cos\theta $$

and

$$ v_{0y}=v_0\sin\theta $$

---

Horizontal Motion

Since horizontal acceleration is zero,

$$ v_x=v_{0x} $$

for all time.

The horizontal position is

$$ x=v_{0x}t $$

---

Vertical Motion

Gravity affects only the vertical direction.

The vertical velocity is

$$ v_y=v_{0y}-gt $$

The vertical position is

$$ y=v_{0y}t-\frac{1}{2}gt^2 $$

---

Time to Reach Maximum Height

At the highest point,

$$ v_y=0 $$

Therefore

$$ 0=v_{0y}-gt $$

Solving for time gives

$$ t=\frac{v_{0y}}{g} $$

---

Maximum Height

The maximum height reached by a projectile is

$$ h=\frac{v_{0y}^2}{2g} $$

---

Range of a Projectile

The horizontal distance traveled before landing is called the range.

The range formula is

$$ R=\frac{v_0^2\sin(2\theta)}{g} $$

This formula is widely used in engineering and ballistics.

---

Relative Velocity

Motion depends on the observer.

Suppose a train moves at

$$ 20\text{ m/s} $$

and a passenger walks inside the train at

$$ 2\text{ m/s} $$

in the same direction.

Relative to the ground, the passenger moves at

$$ 22\text{ m/s} $$

---

General Relative Velocity Formula

The velocity of object A relative to object B is

$$ \vec{v}_{AB}=\vec{v}_A-\vec{v}_B $$

This idea will later become central to Einstein’s theory of relativity.

---

Healthcare Application

Blood flow is fundamentally vector-valued.

Physicians routinely analyze:

• Blood velocity
• Flow direction
• Cardiac output
• Turbulence

Vector analysis helps diagnose:

• Heart valve disorders
• Vessel narrowing
• Circulatory diseases

---

Supply Chain Application

Warehouse automation relies heavily on vectors.

Autonomous robots constantly calculate:

$$ \text{Position} \rightarrow \text{Velocity} \rightarrow \text{Direction} \rightarrow \text{Destination} $$

using vector mathematics.

The same concepts used in projectile motion are used in robotics and logistics.

---

Why This Lesson Matters for Quantum Mechanics

Today we studied vectors in ordinary physical space.

Later, quantum mechanics introduces state vectors.

Instead of describing locations, these vectors describe the state of a quantum system.

Understanding ordinary vectors now makes the transition to quantum mechanics much easier.

---

Key Takeaways

1. Scalars possess magnitude only.
2. Vectors possess magnitude and direction.
3. Vectors can be added and subtracted component-wise.
4. Unit vectors define coordinate directions.
5. Every vector can be decomposed into components.
6. Projectile motion combines horizontal and vertical motion.
7. Gravity affects only the vertical component.
8. Relative velocity depends on the observer.
9. Vectors appear throughout all branches of physics.
10. Quantum states are generalized vectors.

---

References

Young, Hugh D., Freedman, Roger A. (2023). University Physics with Modern Physics (15th Edition). Pearson.

Halliday, David., Resnick, Robert., Walker, Jearl. (2021). Fundamentals of Physics (12th Edition). Wiley.

Serway, Raymond A., Jewett, John W. (2023). Physics for Scientists and Engineers (11th Edition). Cengage Learning.

Taylor, John R. (2004). Classical Mechanics. University Science Books.

Kleppner, Daniel., Kolenkow, Robert J. (2014). An Introduction to Mechanics (2nd Edition). Cambridge University Press.

Feynman, Richard P., Leighton, Robert B., Sands, Matthew. (2011). The Feynman Lectures on Physics, Volume I. Basic Books.

Galileo Galilei. (1638). Discourses and Mathematical Demonstrations Relating to Two New Sciences.

Newton, Isaac. (1687). Philosophiæ Naturalis Principia Mathematica.

---

Next Lesson

Lesson 4: Newton’s Laws of Motion — Why Motion Changes. We will introduce force, inertia, mass, free-body diagrams, Newton’s three laws, and the foundations of classical mechanics.