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:

• 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:

• Inverse trigonometric functions (, , ) 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 :

Quadrants and Signs of Trigonometric Functions

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

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 ( and ) indicate direction along these axes.

• : unit vector in the positive x direction

• : unit vector in the positive y direction

• Vectors can be expressed as

• Example: 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 the vector makes with the x-axis.

• If you know the magnitude and angle :

• If you know the components:

• 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: above x-axis

• Given x and y components:

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: or

• Case 2: Vectors perpendicular to each other

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

• 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:

Worked Example: Vector Addition Using Components

• Given vectors and , resolve each into x and y components:

• Add components:

• Find magnitude and direction:

Summary Table: Vector Addition Cases

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.