Vectors and Coordinate Systems: Study Notes for College Physics

Chapter 2: Vectors

Coordinate Systems

Coordinate systems are essential tools in physics for describing the position of points in space. They provide a framework for specifying locations and directions.

• Definition: A coordinate system consists of a fixed reference point called the origin, specific axes with scales and labels, and instructions for labeling points relative to the origin and axes.

• Types of Coordinate Systems:

  • Cartesian (Rectangular) Coordinate System: Uses perpendicular axes (x, y in 2D; x, y, z in 3D) intersecting at the origin. Points are labeled as (x, y) or (x, y, z).

  • Polar Coordinate System: Specifies a point by its distance r from the origin and angle θ from a reference line (usually the positive x-axis). Points are labeled as (r, θ).

Example: In Cartesian coordinates, the point (3, 8) is 3 units along x and 8 units along y. In polar coordinates, the same point can be described by its distance from the origin and the angle from the x-axis.

Converting Between Coordinate Systems

Conversion between Cartesian and polar coordinates is based on right triangle relationships.

• Polar to Cartesian:

  • $x = r \cos \theta$

  • $y = r \sin \theta$

• Cartesian to Polar:

  • $r = \sqrt{x^2 + y^2}$

  • $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

• Angle θ must be measured counterclockwise from the positive x-axis for these equations to be valid.

Example: The point (-3.50, -2.50) m in Cartesian coordinates converts to polar coordinates as follows:

• $r = \sqrt{(-3.50)^2 + (-2.50)^2} = 4.30\ \text{m}$

• $\theta = \tan^{-1}\left(\frac{-2.50}{-3.50}\right) = 216^\circ$ (quadrant determined by signs)

Vector and Scalar Quantities

Physical quantities in physics are classified as either scalars or vectors.

• Scalar Quantity: Completely specified by a single value with an appropriate unit; has no direction. Examples: Mass, temperature, energy.

• Vector Quantity: Described by both a magnitude (number with units) and a direction. Examples: Displacement, velocity, force.

Some Properties of Vectors

Vectors have unique properties that distinguish them from scalars.

• Notation: Vectors are denoted by boldface letters (A) or with an arrow above (\vec{A}). Magnitude is denoted by italicized letters (A).

• Equality: Two vectors are equal if they have the same magnitude and direction, regardless of their initial points.

• Displacement Example: The displacement vector from point A to B is independent of the path taken; only the initial and final positions matter.

Vector Addition and Subtraction

Vectors can be added or subtracted using graphical or algebraic methods.

• Graphical Addition: Vectors are added "tip-to-tail"; the resultant vector is drawn from the tail of the first to the tip of the last.

• Parallelogram Method: Vectors are placed tail-to-tail; the resultant is the diagonal of the parallelogram formed.

• Commutative Law: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$

• Associative Law: $\vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C}$

• Subtraction: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$; the negative of a vector has the same magnitude but opposite direction.

Multiplying or Dividing a Vector by a Scalar

Multiplying or dividing a vector by a scalar changes its magnitude but not its direction (unless the scalar is negative).

• If the scalar is positive, the direction remains the same.

• If the scalar is negative, the direction is reversed.

Components of a Vector and Unit Vectors

Vectors can be resolved into components along coordinate axes, which simplifies calculations and analysis.

• Component: The projection of a vector along an axis.

• Rectangular Components: For vector A, $A_x$ and $A_y$ are the components along x and y axes, respectively.

• Formulas:

  • $A_x = A \cos \theta$

  • $A_y = A \sin \theta$

  • $A = \sqrt{A_x^2 + A_y^2}$

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

• Unit Vectors: Dimensionless vectors of magnitude 1, used to specify direction. In Cartesian coordinates, i, j, and k are unit vectors along x, y, and z axes, respectively.

• Vector Representation: $\vec{A} = A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k}$

Example: If a hiker walks 25.0 km southeast, the components are $A_x = 25.0 \cos 45^\circ = 17.7$ km, $A_y = 25.0 \sin(-45^\circ) = -17.7$ km.

The Scalar Product of Two Vectors (Dot Product)

The scalar (dot) product combines two vectors to produce a scalar quantity, useful for calculating work and projections.

• Definition: $\vec{A} \cdot \vec{B} = AB \cos \theta$ where θ is the angle between A and B.

• Properties:

  • Dot product is positive if $0^\circ < \theta < 90^\circ$.

  • Zero if $\theta = 90^\circ$ (vectors are perpendicular).

  • Negative if $90^\circ < \theta < 180^\circ$.

• Component Form: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$

Example: If $\vec{A} = (4, 3)$ and $\vec{B} = (5, 2)$, then $\vec{A} \cdot \vec{B} = 4 \times 5 + 3 \times 2 = 20 + 6 = 26$.

The Vector Product of Two Vectors (Cross Product)

The vector (cross) product of two vectors results in a third vector perpendicular to both, with magnitude related to the area of the parallelogram they span.

• Definition: $\vec{A} \times \vec{B} = AB \sin \theta$

• Direction: Given by the right-hand rule: point fingers along A, curl toward B, thumb points in direction of $\vec{A} \times \vec{B}$.

• Anticommutative Property: $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$

• Component Form:

  • $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y) \mathbf{i} + (A_z B_x - A_x B_z) \mathbf{j} + (A_x B_y - A_y B_x) \mathbf{k}$

• Determinant Form:

  • $\vec{A} \times \vec{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$

Example: If $\vec{A} = (2, 3, 4)$ and $\vec{B} = (5, 6, 7)$, then $\vec{A} \times \vec{B}$ can be calculated using the determinant formula above.

Summary Table: Comparison of Scalar and Vector Quantities

Quantity Type	Definition	Examples
Scalar	Magnitude only	Mass, Temperature, Energy
Vector	Magnitude and Direction	Displacement, Velocity, Force

Summary Table: Coordinate System Conversion

From	To	Formula
Polar (r, θ)	Cartesian (x, y)	$x = r \cos \theta$, $y = r \sin \theta$
Cartesian (x, y)	Polar (r, θ)	$r = \sqrt{x^2 + y^2}$, $\theta = \tan^{-1}(y/x)$

Additional info: These notes are based on standard introductory college physics curriculum and expand upon the provided slides and handwritten annotations for completeness and clarity.