Vectors and Trigonometry in Physics: Structured Study Notes

Scalars and Vectors

Definitions and Distinctions

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

• Scalar: A quantity that has only magnitude (size), with no direction. Examples: 5 meters, 20 kilograms, 100 seconds.

• Vector: A quantity that has both magnitude and direction. Examples: 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 adding and subtracting scalars.

  • We will explore vector addition and subtraction in detail later in this chapter.

Trigonometry Review

Pythagorean Theorem

The Pythagorean Theorem describes the relationship between the lengths of the sides of a right triangle.

• Formula: $a^2 + b^2 = c^2$

• Where:

  • a: adjacent side

  • b: opposite side

  • c: hypotenuse

Trigonometry Relationships

Trigonometric functions relate the angles of a triangle to the ratios of its sides. These are essential for resolving vectors into components.

• In a right triangle:

  • $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

  • $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

  • $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

• Inverse trigonometric functions ($\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$) are used to find angles when side lengths are known.

Unit Circle

The unit circle is a circle with a radius of 1. It is useful for understanding the values of trigonometric functions for different angles.

• For a point (x, y) on the unit circle at angle $\theta$:

  • $\sin \theta = y$

  • $\cos \theta = x$

  • $\tan \theta = \frac{y}{x}$

Quadrants and Signs of Trigonometric Functions

The sign of trigonometric functions depends on the quadrant in which the angle lies.

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

Example: In Quadrant II, sine is positive, but cosine and tangent are negative.

Components of Vectors

Vector Components and Unit Vectors

Each vector can be broken down into components along the x and y axes. Unit vectors ($\hat{i}$ and $\hat{j}$) indicate direction along these axes.

• $\hat{i}$: unit vector in the positive x direction

• $\hat{j}$: unit vector in the positive y direction

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

• Example: $10 \hat{i} + 5 \hat{j}$ means the vector points 10 units in x and 5 units in y direction.

Resolving a Vector into x and y Components

To resolve a vector into its components, use trigonometric functions based on the angle $\theta$ the vector makes with the x-axis.

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

  • $A_x = A \cos \theta$

  • $A_y = A \sin \theta$

• If you know the components:

  • $A = \sqrt{A_x^2 + A_y^2}$

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

• Common Mistake: The x component does not always use the cosine function; it depends on whether the angle is adjacent to the x-axis.

Systems for Defining a Vector

• Given magnitude and direction: $A = 13 \text{ m at } 67^\circ$ above x-axis

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

Vector Addition and Subtraction

Methods for Adding Vectors

There are three main cases for vector addition and subtraction, depending on their relative directions.

• Case 1: Vectors along the same direction

  • Add/subtract like scalars: $A + B$ or $A - B$

• Case 2: Vectors perpendicular to each other

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

  • $C = \sqrt{A^2 + B^2}$

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

• 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 tangent to find magnitude and direction.

Component Method Steps

1. Resolve all vectors into x and y components.

2. Add all x components to get the resultant x component; add all y components to get the resultant y component.

3. Find the resultant vector's magnitude and direction:

   • $C_x = A_x + B_x$

   • $C_y = A_y + B_y$

   • $C = \sqrt{C_x^2 + C_y^2}$

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

Worked Example: Vector Addition Using Components

• Given vectors $\vec{A}$ and $\vec{B}$, resolve each into x and y components:

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

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

• Add components:

  • $C_x = A_x + B_x$

  • $C_y = A_y + B_y$

• Find magnitude and direction:

  • $C = \sqrt{C_x^2 + C_y^2}$

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

Summary Table: Vector Addition Cases

Case	Addition	Subtraction	Method
Same Direction	$A + B$	$A - B$	Scalar addition/subtraction
Perpendicular	$C = \sqrt{A^2 + B^2}$	$C = \sqrt{A^2 + B^2}$	Pythagorean Theorem
Not Parallel/Perpendicular	Component method	Component method	Resolve into components, then add

Key Takeaways

• Always distinguish between scalars and vectors in physics problems.

• Use trigonometric relationships to resolve vectors into components.

• Apply the correct method for vector addition based on their relative directions.

• Component method is universally applicable for any vector directions.

Additional info: Some explanations and examples have been expanded for clarity and completeness beyond the original notes.