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 many other physical phenomena.

• Scalar Quantity: A quantity described by a single number (magnitude) only, 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 typically denoted with an arrow above the letter, e.g., $\vec{A}$.

Example: Displacement, velocity, and acceleration are all vector quantities.

Properties of Vectors

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

• Magnitude: The length of the vector, representing its size.

• Direction: The orientation of the vector in space.

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

Vector Addition and Subtraction

Vectors can be added or subtracted to find resultant quantities.

• Tip-to-Tail Method: 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.

• Parallelogram Rule: Place both vectors with their tails at the same point; the diagonal of the parallelogram formed is the resultant.

• 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): $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$.

Example: A hiker walks 4 miles east, then 3 miles north. The net displacement is found using the Pythagorean theorem:

• $d = \sqrt{4^2 + 3^2} = 5$ miles

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

Vector Components and Decomposition

Any vector in a plane can be decomposed into components along the axes of a coordinate system. This simplifies calculations and allows for algebraic manipulation.

• Component Vectors: The projections of a vector along the coordinate axes (usually x and y).

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

• Finding Components: If a vector $\vec{A}$ has magnitude $A$ and makes an angle $\theta$ with the x-axis:

• $A_x = A \cos \theta$

• $A_y = A \sin \theta$

Note: If the vector points left or down, the corresponding component is negative. Always check the direction before assigning signs.

Example: A vector of 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 with magnitude 1 and no units, used to specify directions along coordinate axes.

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

• 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

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

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

• Scalar Multiplication: Multiply each component by the scalar: $c\vec{A} = (cA_x) \hat{i} + (cA_y) \hat{j}$

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

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

Changing Between Geometric and Component Representations

• To decompose a vector: Use trigonometry to find components from magnitude and angle.

• To reassemble a vector: Use the Pythagorean theorem and inverse tangent to find magnitude and direction from components.

• Be careful with signs: Assign negative signs to components pointing left (negative x) or down (negative y).

Tilted Axes and Arbitrary Directions

Sometimes, it is convenient to choose axes that are not aligned with the standard horizontal and vertical directions, especially when analyzing forces or motion along inclined planes or surfaces.

• Axes remain perpendicular but can be oriented to match the problem's geometry.

• Components are then found parallel and perpendicular to the chosen axes.

Example: Decomposing a muscle force into components parallel and perpendicular to a bone using axes aligned with the bone.

Summary Table: Vector Operations

Operation	Geometric Method	Algebraic Method
Addition	Tip-to-tail or 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
Magnitude	Length of arrow	$\sqrt{A_x^2 + A_y^2}$
Direction	Angle with respect to axis	$\tan^{-1}(A_y / A_x)$

Key Formulas

• Components: $A_x = A \cos \theta$, $A_y = A \sin \theta$

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

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

• Vector in Unit Vector Notation: $\vec{A} = A_x \hat{i} + A_y \hat{j}$

Applications of Vectors in Physics

• Vectors are used to describe position, displacement, velocity, acceleration, force, momentum, electric and magnetic fields, and more.

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

Additional info: These notes expand on the textbook's brief points to provide a self-contained summary suitable for exam preparation, including explicit formulas, examples, and a summary table for vector operations.