Vectors and Coordinate Systems: Study Notes

Vectors and Coordinate Systems

Introduction to Vectors

In physics, many quantities require both a magnitude (size) and a direction for their complete description. Such quantities are called vectors. Understanding vectors and their mathematical manipulation is essential for analyzing motion, forces, and other physical phenomena.

• Scalar Quantity: A quantity described by a single number (magnitude) and units, such as mass, temperature, or volume.

• Vector Quantity: A quantity described by both magnitude and direction, such as displacement, velocity, acceleration, force, and momentum.

• Geometric Representation: Vectors are represented as arrows; the length indicates magnitude, and the arrowhead indicates direction.

• Notation: Vectors are denoted with an arrow above the symbol, e.g., $\vec{v}$ for velocity.

Properties of Vectors

Vectors are defined by their magnitude and direction, regardless of their initial position. Two vectors are equal if they have the same magnitude and direction, even if they start at different points.

• Displacement Example: If two people walk 200 ft northeast from different starting points, their displacement vectors are equal.

• Equality of Vectors: $\vec{A} = \vec{B}$ if $|\vec{A}| = |\vec{B}|$ and both point in the same direction.

Vector Addition and Subtraction

Vectors can be added or subtracted using graphical or algebraic methods. The most common graphical method is the "tip-to-tail" rule.

• Tip-to-Tail Rule: Place the tail of the second vector at the tip of the first; the resultant vector is drawn from the tail of the first to the tip of the last.

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

• Subtraction: To subtract $\vec{B}$ from $\vec{A}$, add $-\vec{B}$ (same magnitude, opposite direction): $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$.

• Parallelogram Rule: When two vectors are drawn from the same point, the diagonal of the parallelogram they form is their sum.

Example: A hiker walks 4 miles east, then 3 miles north. The net displacement $\vec{D}$ is:

• Magnitude: $|\vec{D}| = \sqrt{4^2 + 3^2} = 5$ miles

• Direction: $\theta = \tan^{-1}(3/4) = 36.9^\circ$ north of east

Adding More Than Two Vectors

Vector addition extends to any number of vectors by repeated application of the tip-to-tail method or by adding all components algebraically.

• Net Displacement: The sum of several displacement vectors gives the overall change in position.

Coordinate Systems

A coordinate system is a grid imposed on a problem to define positions and directions. The most common is the Cartesian (x, y) system, but axes can be tilted for convenience.

• Origin: The reference point (0,0).

• Axes Orientation: Axes can be oriented as needed; for example, parallel and perpendicular to a surface.

• Quadrants: The x-y plane is divided into four quadrants.

• Application: GPS uses Earth's coordinate system to determine position.

Vector Components and Decomposition

Any vector can be decomposed into components parallel to the coordinate axes. This simplifies calculations and allows for algebraic manipulation.

• Component Vectors: For vector $\vec{A}$, the components are $A_x$ (along x-axis) and $A_y$ (along y-axis).

• Decomposition: $\vec{A} = A_x \hat{i} + A_y \hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors in the x and y directions, respectively.

• Magnitude and Direction:

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

  • Direction: $\theta = \tan^{-1}(A_y / A_x)$

• Signs: If a component points left (negative x) or down (negative y), a minus sign must be included.

Example: A vector 5 units at 30° above the x-axis has components:

• $A_x = 5 \cos 30^\circ = 4.33$

• $A_y = 5 \sin 30^\circ = 2.5$

Unit Vectors

Unit vectors are vectors of magnitude 1 with no units, used to specify direction along coordinate axes.

• Symbols: $\hat{i}$ (x-direction), $\hat{j}$ (y-direction), $\hat{k}$ (z-direction in 3D)

• Usage: Any vector can be written as a sum of its components times the corresponding unit vectors.

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

Algebraic Vector Operations

Vector addition, subtraction, and scalar multiplication can be performed algebraically by operating on components.

• Addition: $\vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$

• Subtraction: $\vec{A} - \vec{B} = (A_x - B_x)\hat{i} + (A_y - B_y)\hat{j}$

• Scalar Multiplication: $c\vec{A} = (cA_x)\hat{i} + (cA_y)\hat{j}$

Moving Between Geometric and Component Representations

Vectors can be described either by their magnitude and direction (geometric) or by their components (algebraic). Conversion between these forms is essential for problem-solving.

• From Components to Magnitude and Direction:

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

  • Direction: $\theta = \tan^{-1}(A_y / A_x)$

• From Magnitude and Direction to Components:

  • $A_x = |\vec{A}| \cos \theta$

  • $A_y = |\vec{A}| \sin \theta$

• Sign Convention: Insert minus signs for components pointing left or down.

Tilted Axes and Arbitrary Directions

Sometimes, it is convenient to tilt the coordinate axes to align with a surface or direction of interest. The process of decomposition remains the same, but the axes are redefined.

• Application: Decomposing forces parallel and perpendicular to a bone or inclined plane.

• Example: A muscle pulls at 15° to the bone; components are found using $F_{\parallel} = F \cos 15^\circ$, $F_{\perp} = F \sin 15^\circ$.

Summary Table: Vector Operations

Operation	Graphical Method	Algebraic Method
Addition	Tip-to-tail, parallelogram	Add components: $A_x + B_x$, $A_y + B_y$
Subtraction	Add the negative vector	Subtract components: $A_x - B_x$, $A_y - B_y$
Scalar Multiplication	Change length, keep direction (or reverse if negative)	Multiply each component by scalar: $cA_x$, $cA_y$
Magnitude	Measure length	$|\vec{A}| = \sqrt{A_x^2 + A_y^2}$
Direction	Measure angle from axis	$\theta = \tan^{-1}(A_y / A_x)$

Key Formulas

• Magnitude of a Vector: $|\vec{A}| = \sqrt{A_x^2 + A_y^2}$

• Direction (angle from x-axis): $\theta = \tan^{-1}(A_y / A_x)$

• Component Form: $\vec{A} = A_x \hat{i} + A_y \hat{j}$

• From Magnitude and Angle:

  • $A_x = |\vec{A}| \cos \theta$

  • $A_y = |\vec{A}| \sin \theta$

Applications of Vectors

• Vectors are used throughout physics: in describing motion (velocity, acceleration), forces, electric and magnetic fields, and more.

• Mastery of vector operations is foundational for all subsequent topics in physics with calculus.

Additional info:

• In three dimensions, vectors have a z-component and use $\hat{k}$ as the unit vector in the z-direction.

• Vector operations extend naturally to three dimensions using the same principles.