Vectors and Coordinate Systems: Study Notes

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Vectors and Coordinate Systems

Review of Vectors and Scalars

Physical quantities in physics are classified as either scalars or vectors. Scalars have only magnitude (size), while vectors have both magnitude and direction.

Scalar: A quantity described by magnitude alone (e.g., temperature, distance, speed).
Vector: A quantity described by both magnitude and direction (e.g., force, displacement, velocity).

Examples:

Measurement	Quantity	Magnitude?	Direction?	Vector/Scalar
"It’s 60°F outside"	Temperature	Yes	No	Scalar
"I pushed with 100N north"	Force	Yes	Yes	Vector
"I walked for 10 m"	Distance	Yes	No	Scalar
"I walked 10 m east"	Displacement	Yes	Yes	Vector
"I drove at 80 mph"	Speed	Yes	No	Scalar
"I drove 80 mph west"	Velocity	Yes	Yes	Vector

Introduction to Vector Math

Unlike scalars, vectors require special rules for addition and subtraction due to their directional nature. Vectors are typically represented as arrows, where the length indicates magnitude and the arrowhead indicates direction.

Adding Scalars: Simple arithmetic (e.g., 3 kg + 4 kg = 7 kg).
Adding Perpendicular Vectors: Use the Pythagorean theorem to find the resultant (e.g., walking 3 m right and 4 m up forms a right triangle; resultant = 5 m).
Adding Parallel Vectors: Add magnitudes directly if in the same direction.

Example: If you walk 10 m to the right and then 6 m to the left, your total displacement is 4 m to the right.

Adding Vectors Graphically

Vectors are added graphically by placing them tip-to-tail. The resultant vector (often labeled as $ \vec{C} $ or $ \vec{R} $) is the shortest path from the start of the first vector to the end of the last vector. Vector addition is commutative: $ \vec{A} + \vec{B} = \vec{B} + \vec{A} $.

Resultant Vector: The vector sum of two or more vectors.
Tip-to-Tail Method: Place the tail of the next vector at the tip of the previous vector.

Formula for Perpendicular Vectors:

$\vec{R} = \sqrt{A^2 + B^2}$

Example: A delivery truck travels 8 miles in the +x direction, 5 miles in the +y direction, and 4 miles again in the +x direction. The total displacement is found by summing the x and y components and applying the Pythagorean theorem.

Subtracting Vectors Graphically

Subtracting vectors is similar to adding, but you reverse the direction of the vector being subtracted. The order of subtraction matters: $ \vec{A} - \vec{B} \neq \vec{B} - \vec{A} $.

Negative Vector: Same magnitude, opposite direction.
Order Matters: Subtraction is not commutative.

Formula:

$\vec{C} = \vec{A} - \vec{B}$

Adding Multiples of Vectors

Multiplying a vector by a scalar changes its magnitude but not its direction. If the scalar is negative, the direction is reversed.

Multiplying by a number greater than 1 increases the magnitude.
Multiplying by a number between 0 and 1 decreases the magnitude.

Example: $ \vec{C} = 3\vec{A} - 2\vec{B} $

Vector Composition and Decomposition

Any vector can be broken down into components along the x and y axes (decomposition), or constructed from its components (composition).

Decomposition: $ A_x = A \cos(\theta_x) $, $ A_y = A \sin(\theta_x) $
Composition: $ A = \sqrt{A_x^2 + A_y^2} $, $ \theta_x = \tan^{-1}(A_y/A_x) $

Example: If $ A_x = 8 $ m and $ A_y = 6 $ m, then $ A = 10 $ m and $ \theta_x = 36.9^\circ $.

Vector Addition by Components

Vectors can be added by summing their respective x and y components, then finding the magnitude and direction of the resultant.

Sum all x-components: $ R_x = A_x + B_x + ... $
Sum all y-components: $ R_y = A_y + B_y + ... $
Resultant magnitude: $ R = \sqrt{R_x^2 + R_y^2} $
Resultant direction: $ \theta_R = \tan^{-1}(R_y/R_x) $

Example: You walk 5 m at 53° above the +x-axis, then 8 m at 30° above the +x-axis. Calculate the total displacement using components.

Vectors in Any Quadrant

Vectors can point in any direction, so their components may be positive or negative depending on the quadrant. The reference angle is always measured from the nearest x-axis.

Magnitudes are always positive; components can be negative.
Use the signs of the components to determine the vector's quadrant.
To find the absolute angle from the +x-axis, use the reference angle and adjust based on the quadrant.

Formulas:

$A_x = A \cos(\theta_x)$ $A_y = A \sin(\theta_x)$ $A = \sqrt{A_x^2 + A_y^2}$ $\theta_x = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

Describing Vector Directions with Words

Directions can be described using angles (counterclockwise is positive, clockwise is negative) or compass points (e.g., "30° north of east"). The reference angle for component equations is always positive and measured from the nearest x-axis.

"30° north of east" means start east, then rotate 30° towards north.
"53° west of south" means start south, then rotate 53° towards west.

Unit Vectors

Unit vectors are special vectors with magnitude 1, used to indicate direction along coordinate axes. In two dimensions:

$ \hat{i} $: Points in the +x direction
$ \hat{j} $: Points in the +y direction
$ \hat{k} $: Points in the +z direction (for 3D)

Any vector can be written as a sum of its components multiplied by unit vectors:

$\vec{A} = A_x \hat{i} + A_y \hat{j}$

Example: $ \vec{A} = 4\hat{i} + 2\hat{j} $, $ \vec{B} = -\hat{i} + 2\hat{j} $, $ \vec{R} = \vec{A} + \vec{B} = 3\hat{i} + 4\hat{j} $

Practice Problems and Applications

Calculate the magnitude and direction of a vector given its components.
Express vectors in unit vector notation and find the resultant of multiple vectors.
Apply vector addition and subtraction in real-world contexts (e.g., displacement, force).

Example: Given $ \vec{A} = (3\hat{i} - 3\hat{j}) $ m, $ \vec{B} = (\hat{i} - 4\hat{j}) $ m, and $ \vec{C} = (-2\hat{i} + 5\hat{j}) $ m, find the magnitude and direction of $ \vec{D} = \vec{A} + \vec{B} + \vec{C} $.

Additional info: These notes cover all foundational aspects of vectors and coordinate systems, including graphical and analytical methods, vector decomposition, and the use of unit vectors. Mastery of these concepts is essential for further study in physics, especially in mechanics and electromagnetism.