Vectors in Physics

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Review of Vectors and Scalars

In physics, quantities are classified as either vectors or scalars. Understanding the distinction is essential for analyzing physical phenomena.

• Scalar: A quantity described only by its magnitude (size). Examples: temperature, distance, speed.

• Vector: A quantity described by both magnitude and direction. Examples: force, displacement, velocity.

Measurements with direction are vectors; without direction, they are scalars.

Intro to Vector Math

Vectors are added and subtracted differently than scalars due to their directional nature.

• Combining Scalars: Simple addition (e.g., 3 kg + 4 kg = 7 kg).

• Combining Parallel Vectors: Add just like normal numbers if they point in the same direction.

• Combining Perpendicular Vectors: Use the Pythagorean theorem (triangle math):

Example: If you walk 5 m to the right, then 5 m up, your total displacement is m.

Adding Vectors Graphically

Vectors are represented as arrows. The resultant vector is the shortest path from the start of the first vector to the end of the last.

• Tip-to-tail method: Place the tail of the next vector at the tip of the previous one.

• Order does not matter when adding vectors.

Example: For vectors and , the resultant is .

Subtracting Vectors Graphically

Subtracting vectors is similar to addition, but you reverse the direction of the vector being subtracted.

• Negative vector: Same magnitude, opposite direction.

• When subtracting, order does matter.

Example:

Adding Multiples of Vectors

Multiplying a vector by a scalar changes its magnitude but not its direction.

• If scalar , magnitude increases; if , magnitude decreases.

• Negative scalars reverse direction.

Example:

Vector Composition and Decomposition

Vectors can be broken into components or composed from components using trigonometry.

• Composition: Combine and components to get magnitude and direction.

• Decomposition: Use and to find components from magnitude and angle.

Formulas:

Vector Addition by Components

To add vectors, sum their and components separately.

• Resultant magnitude:

• Resultant direction:

Vectors in All Quadrants

When working in all quadrants, pay attention to the signs of components and the reference angle.

• Use the correct sign for and components based on the quadrant.

• Absolute angle: , then adjust for quadrant.

Describing Directions with Words

Directions may be given as angles from axes or as compass directions (e.g., "30° south of east").

• Convert compass directions to standard angles for calculation.

• Calculate components using and with the given angle.

Unit Vectors

Unit vectors are vectors of magnitude 1, used to specify direction in component form.

• : unit vector in -direction

• : unit vector in -direction

• : unit vector in -direction

• Any vector:

Dot Product (Scalar Product)

The dot product of two vectors produces a scalar and measures how much one vector extends in the direction of another.

• Using components:

• Zero if vectors are perpendicular.

Cross Product (Vector Product) and the Right-Hand Rule

The cross product of two vectors produces a vector perpendicular to both, with magnitude:

• Direction given by the right-hand rule.

• Component form:

Summary Table: Vector Operations

Practice Problems and Examples

• Calculate the magnitude and direction of a resultant vector given its components.

• Express vectors in unit vector notation.

• Find the dot and cross products of given vectors.

• Determine the angle between two vectors given their dot and cross products.

Example: Given and , find .

Additional info: These notes cover all foundational aspects of vectors in physics, including graphical and analytical methods, and provide practice for mastery.