Introduction
In Lesson 2, we studied motion along a straight line. Such motion is called one-dimensional motion because only one coordinate is needed to describe an object’s position.
However, most motion in nature occurs in two or three dimensions.
Examples include:
- A soccer ball flying through the air.
- An airplane traveling across a continent.
- Blood flowing through branching arteries.
- A warehouse robot navigating shelves.
- Earth orbiting the Sun.
To describe these motions mathematically, we need a new concept called a vector.
Vectors are among the most important ideas in all of physics. They appear in:
- Classical mechanics
- Electromagnetism
- Fluid mechanics
- Quantum mechanics
- General relativity
- String theory
In fact, quantum states themselves will eventually be represented as vectors in abstract spaces called Hilbert spaces.
Before reaching those advanced topics, we must first understand ordinary vectors in physical space.
Scalars and Vectors
Physical quantities can be divided into two major categories.
Scalars
A scalar has magnitude only.
Examples include:
- Mass
- Time
- Temperature
- Energy
- Distance
- Speed
Suppose a car travels at:
$$ 60 \text{ km/h} $$
This tells us how fast the car is moving.
It does not tell us the direction.
Therefore speed is a scalar.
Vectors
A vector has:
- Magnitude
- Direction
Examples include:
- Velocity
- Acceleration
- Force
- Momentum
- Electric field
Suppose a car travels:
$$ 60 \text{ km/h East} $$
Now both magnitude and direction are specified.
Therefore velocity is a vector.
Representing Vectors
Vectors are usually written with arrows.
Examples:
$$ \vec{A} $$
$$ \vec{v} $$
$$ \vec{F} $$
The magnitude of a vector is written as:
$$ |\vec{A}| $$
The magnitude represents the length of the vector.
Components of a Vector
Suppose a vector points northeast.
Instead of treating it as a single object, we can break it into:
- Horizontal component
- Vertical component
Suppose:
$$ \vec{A}=(3,4) $$
This means:
- 3 units in the x-direction
- 4 units in the y-direction
Magnitude of a Vector
The magnitude of a vector follows directly from the Pythagorean theorem.
For a vector
$$ \vec{A}=(A_x,A_y) $$
the magnitude is
$$ |\vec{A}|=\sqrt{A_x^2+A_y^2} $$
Example
Consider:
$$ \vec{A}=(3,4) $$
The magnitude becomes
$$ |\vec{A}|=\sqrt{3^2+4^2} $$
$$ |\vec{A}|=\sqrt{9+16} $$
$$ |\vec{A}|=5 $$
Thus the vector has length 5 units.
Direction of a Vector
The direction angle is determined using trigonometry.
Suppose a vector has components:
$$ A_x $$
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
$$ \theta=\tan^{-1}\left(\frac{4}{3}\right) $$
$$ \theta=53.13^\circ $$
Unit Vectors
A unit vector is a vector with magnitude one.
The standard unit vectors are:
Horizontal direction:
$$ \hat{i} $$
Vertical direction:
$$ \hat{j} $$
Their magnitudes satisfy:
$$ |\hat{i}|=1 $$
$$ |\hat{j}|=1 $$
Writing Vectors Using Unit Vectors
Instead of writing
$$ \vec{A}=(3,4) $$
we may write
$$ \vec{A}=3\hat{i}+4\hat{j} $$
This notation is extremely common in physics.
Vector Addition
Suppose
$$ \vec{A}=(3,4) $$
and
$$ \vec{B}=(2,1) $$
Then
$$ \vec{A}+\vec{B}=(3+2,4+1) $$
Therefore
$$ \vec{A}+\vec{B}=(5,5) $$
Vector addition is performed component by component.
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
$$ \theta $$
The horizontal component is
$$ A_x=A\cos\theta $$
The 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 Vector
The position of an object in two dimensions is described by a position vector.
The position vector is
$$ \vec{r}=x\hat{i}+y\hat{j} $$
where:
- x is the horizontal position
- y is the vertical position
This is one of the most important equations in introductory mechanics.
Velocity in Two Dimensions
Velocity is also a vector.
The velocity vector is
$$ \vec{v}=v_x\hat{i}+v_y\hat{j} $$
The speed is its magnitude:
$$ |\vec{v}|=\sqrt{v_x^2+v_y^2} $$
Acceleration in Two Dimensions
Acceleration is similarly written as
$$ \vec{a}=a_x\hat{i}+a_y\hat{j} $$
with magnitude
$$ |\vec{a}|=\sqrt{a_x^2+a_y^2} $$
This gets us to the vector foundation. In the next part of Lesson 3, we would continue with:
- Projectile Motion
- Launch Angle
- Time of Flight
- Maximum Height
- Range Formula
- Relative Velocity
- Healthcare Applications
- Supply Chain Applications
- Connection to Quantum Mechanics
in the same copy-ready format with every equation on a single line between $$ ... $$. This format should paste cleanly into WordPress with MathJax/KaTeX.
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