Vectors and Vector Operations (Ch. 3) - Study Notes

Vectors and Vector Operations

Introduction to Scalars and Vectors

In physics, quantities are classified as either scalars or vectors. Understanding the distinction is essential for analyzing physical phenomena.

• Scalars: Have magnitude only and follow standard algebraic rules. Examples: mass, time, pressure, temperature, energy, power.

• Vectors: Have both magnitude and direction, and require special rules for combination. Examples: displacement, velocity, acceleration, force, momentum.

Representation of Vectors

Vectors are typically represented in boldface (e.g., A) or with an arrow above the letter (e.g., $\vec{A}$). The direction is indicated by the arrow, and the length represents the magnitude.

• Tail: The starting point of the vector.

• Tip (Head): The ending point, indicating direction.

Preliminaries: Angles and Basic Trigonometry

Angles are measured relative to the positive x-axis. They can be expressed in degrees or radians, and trigonometric functions are used to resolve vector components.

• Conversion: $\text{Angle in radians} = \frac{\text{Angle in degrees}}{180} \pi$

• Trigonometric functions:

  • $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

  • $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

  • $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

• Pythagorean theorem: $x^2 + y^2 = r^2$

Quadrants:

• Quadrant 1: $0^\circ$ – $90^\circ$

• Quadrant 2: $90^\circ$ – $180^\circ$

• Quadrant 3: $180^\circ$ – $270^\circ$

• Quadrant 4: $270^\circ$ – $360^\circ$

Inverse trigonometric functions are used to find angles from known ratios:

• Example: $\sin \theta = 0.866 \Rightarrow \theta = \sin^{-1}(0.866) = 60.0^\circ$

• Example: $\tan^{-1}(0.400) = 21.8^\circ$

Equal Vectors

Two vectors $\vec{A}$ and $\vec{B}$ are equal if they have the same magnitude and same direction. Their location in space does not affect equality.

• Moving a vector does not change its magnitude or direction.

Vector Components and Unit Vectors

A unit vector has a magnitude of 1 and points in a specific direction. Unit vectors are used to specify directions along coordinate axes:

• i: x-direction

• j: y-direction

• k: z-direction

Any vector $\vec{a}$ in 3D can be written as:

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

• Where $a_x$, $a_y$, $a_z$ are the scalar components along each axis.

For a 2D vector $\vec{a}$:

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

• $a_x = a \cos(\theta)$

• $a_y = a \sin(\theta)$

Vector Addition and Subtraction

By Components (Analytical/Algebraic Method)

Vectors can be added or subtracted by combining their respective components:

• Addition: $\vec{r} = \vec{a} + \vec{b} + \vec{c}$

• Subtraction: $\vec{d} = \vec{a} - \vec{b}$

• Component-wise: $r_x = a_x + b_x + c_x$, $r_y = a_y + b_y + c_y$

• Magnitude: $r = \sqrt{r_x^2 + r_y^2}$

• Direction: $\theta = \tan^{-1}\left(\frac{r_y}{r_x}\right)$

Geometric Methods

• Tip-to-Tail Method: Place the tail of the next vector at the tip of the previous. The resultant is drawn from the tail of the first to the tip of the last. $\vec{R} = \vec{A} + \vec{B}$

• Parallelogram Method: Vectors are placed tail-to-tail. A parallelogram is formed, and the diagonal represents the resultant.

Properties:

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

• Vector addition is associative: $\vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C}$

Vector Multiplication

Multiplying a Vector by a Scalar

Multiplying a vector $\vec{a}$ by a scalar $s$ scales its magnitude but does not change its direction (unless $s$ is negative, which reverses direction).

• By components: $s\vec{a} = sa_x \hat{i} + sa_y \hat{j} + sa_z \hat{k}$

Dot Product (Scalar Product)

The dot product of two vectors yields a scalar and measures how much one vector extends in the direction of another.

• $\vec{a} \cdot \vec{b} = ab \cos(\phi)$, where $\phi$ is the angle between $\vec{a}$ and $\vec{b}$

• In unit-vector notation: $\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z$

• Dot product is commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$

Cross Product (Vector Product)

The cross product of two vectors yields a vector that is perpendicular to the plane defined by the original vectors.

• $\vec{a} \times \vec{b} = \vec{c}$

• Magnitude: $c = ab \sin(\phi)$, where $\phi$ is the smaller angle between $\vec{a}$ and $\vec{b}$

• Direction: Perpendicular to the plane of $\vec{a}$ and $\vec{b}$, given by the right-hand rule

• Cross product is not commutative: $\vec{a} \times \vec{b} = - (\vec{b} \times \vec{a})$

In unit-vector notation:

• $\vec{a} \times \vec{b} = (a_y b_z - a_z b_y)\hat{i} + (a_z b_x - a_x b_z)\hat{j} + (a_x b_y - a_y b_x)\hat{k}$

Right-Hand Rule for Cross Product

To determine the direction of the cross product, point your right hand's fingers in the direction of the first vector, curl them toward the second vector, and your thumb points in the direction of the resultant vector.

Summary Table: Vector Operations

Operation	Result	Formula	Commutative?
Scalar Multiplication	Vector	$s\vec{a}$	Yes
Dot Product	Scalar	$\vec{a} \cdot \vec{b} = ab \cos(\phi)$	Yes
Cross Product	Vector	$\vec{a} \times \vec{b} = ab \sin(\phi)$	No

Examples

• Dot product of unit vectors: $\hat{i} \cdot \hat{j} = 0$, $\hat{i} \cdot \hat{i} = 1$

• Cross product of unit vectors: $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$

• Other examples: See class notes for specific applications in physics problems.

Additional info: These notes provide a comprehensive overview of vector operations, which are foundational for topics such as kinematics, forces, and more advanced physics concepts.