Vectors and Trigonometry in Physics: Structured Study Notes

Scalars and Vectors

Definitions and Distinctions

In physics, quantities are classified as either scalars or vectors. Understanding the difference is essential for solving problems involving direction and magnitude.

• Scalar: A quantity with only magnitude (numerical value), no direction. Example: 5 meters, 20 seconds.

• Vector: A quantity with both magnitude and direction. Example: 5 meters North, 6 m/s Southwest.

• Why does it matter if a quantity is a scalar or vector? Adding and subtracting vectors is very different from scalars due to the directional component. This affects how you solve physics problems.

Trigonometry Review

Pythagorean Theorem

The Pythagorean Theorem relates the lengths of the sides of a right triangle:

• a: adjacent side

• b: opposite side

• c: hypotenuse

Equation:

$a^2 + b^2 = c^2$

Trigonometry Relationships

Trigonometric functions relate the angles and sides of a right triangle:

• $\cos \theta = \frac{\text{adj}}{\text{hyp}}$

• $\sin \theta = \frac{\text{opp}}{\text{hyp}}$

• $\tan \theta = \frac{\text{opp}}{\text{adj}}$

Inverse trig functions help you find the angle when you know the side lengths:

• $\theta = \cos^{-1}\left(\frac{\text{adj}}{\text{hyp}}\right)$

• $\theta = \sin^{-1}\left(\frac{\text{opp}}{\text{hyp}}\right)$

• $\theta = \tan^{-1}\left(\frac{\text{opp}}{\text{adj}}\right)$

Unit Circle

The unit circle (radius = 1) helps visualize and calculate trigonometric functions for any angle.

• $\sin \theta = \frac{y}{r}$

• $\cos \theta = \frac{x}{r}$

• $\tan \theta = \frac{y}{x}$

Quadrants and Trigonometric Signs

Trigonometric functions change sign depending on the quadrant:

Quadrant	Angle Range	Functions Positive
Quadrant I	0°–90°	All
Quadrant II	90°–180°	Sin
Quadrant III	180°–270°	Tan
Quadrant IV	270°–360°	Cos

Knowing the quadrant helps determine the sign of the trigonometric value.

Components of Vectors

Vector Components

Each vector can be broken into x and y components, representing its magnitude in each direction.

• Unit vectors: $\hat{i}$ (x-direction), $\hat{j}$ (y-direction)

• Vectors can be expressed as $A = A_x \hat{i} + A_y \hat{j}$

• Example: $10 \hat{i} + 5 \hat{j}$ means 10 units in x, 5 units in y.

Resolving Vectors into Components

To resolve a vector into x and y components, use trigonometric functions:

• If you know the angle $\theta$ and magnitude $A$:

  • $A_x = A \cos \theta$

  • $A_y = A \sin \theta$

• Be careful: The x component does not always use cosine; it depends on the triangle's orientation.

Common Mistake: Always check if the side is adjacent or opposite to the angle before applying sine or cosine.

Systems for Defining Vectors

• Given magnitude and direction: Example: 13 m at 67° above x-axis.

• Given x and y components: Example: $A = 5 \hat{i} + 12 \hat{j}$

Adding and Subtracting Vectors

Methods for Vector Addition/Subtraction

There are three main cases for adding/subtracting vectors:

• Case 1: Vectors along the same direction

  • Add/subtract like scalars.

• Case 2: Vectors perpendicular to each other

  • Add head-to-tail and use Pythagorean Theorem.

  • $C = \sqrt{A^2 + B^2}$

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

• Case 3: Vectors not parallel or perpendicular

  • Use the component method: resolve each vector into x and y components, add components, then use Pythagorean Theorem and inverse trig functions.

Component Method Steps

1. Resolve all vectors into x and y components.

2. Add all x components together for the resultant x component; add all y components for the resultant y component.

3. Find the resultant vector's magnitude and direction using Pythagorean Theorem and inverse tangent.

Example for vectors $\vec{A}$ and $\vec{B}$:

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

• $\vec{B} = B_x \hat{i} + B_y \hat{j}$

• $C_x = A_x + B_x$

• $C_y = A_y + B_y$

• $C = \sqrt{C_x^2 + C_y^2}$

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

Summary Table: Vector Addition Cases

Case	Addition	Subtraction	Method
Same Direction	Scalar addition	Scalar subtraction	Simple arithmetic
Perpendicular	Head-to-tail, Pythagorean	Head-to-tail, Pythagorean	Use $a^2 + b^2 = c^2$
Not Parallel/Perpendicular	Component method	Component method	Resolve into x/y, then add

Key Points and Applications

• Vectors are essential in physics for describing quantities like displacement, velocity, and force.

• Trigonometry is used to resolve vectors and calculate magnitudes and directions.

• Component method is a universal approach for vector addition/subtraction.

• Always check the orientation and quadrant when using trigonometric functions.

Example Application

Suppose you have two displacement vectors: 5 m east and 12 m north. To find the resultant displacement:

• Resolve into components: $A_x = 5$, $A_y = 0$; $B_x = 0$, $B_y = 12$

• Add components: $C_x = 5$, $C_y = 12$

• Magnitude: $C = \sqrt{5^2 + 12^2} = 13$

• Direction: $\theta = \tan^{-1}(12/5) \approx 67.4^\circ$ north of east

Additional info: These notes expand on the original slides by providing full definitions, step-by-step methods, and example calculations for vector operations and trigonometric relationships in physics.