Vectors and Scalars

Definitions and Distinctions

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

• Scalar: A quantity with only magnitude (size), such as temperature or mass.

• Vector: A quantity with both magnitude and direction, such as force or velocity.

Examples:

• Temperature: 60°F (Scalar)

• Force: 100 N north (Vector)

• Distance: 10 m (Scalar)

• Displacement: 10 m east (Vector)

• Speed: 80 mph (Scalar)

• Velocity: 80 mph west (Vector)

Vector Representation and Basic Operations

Graphical Representation

Vectors are represented as arrows. The length of the arrow indicates the magnitude, and the direction of the arrow shows the direction of the vector.

Adding and Subtracting Vectors

Vector addition and subtraction are performed using the tip-to-tail method. The resultant vector is the shortest path from the start of the first vector to the end of the last vector.

• Commutative Property: The order of addition does not affect the result:

• Subtraction: Subtracting a vector is equivalent to adding its negative (same magnitude, opposite direction).

Example: If you walk 3 m right and then 4 m up, your total displacement forms a right triangle. The magnitude is found using the Pythagorean theorem:

Combining Vectors

• Parallel Vectors: Add magnitudes directly if in the same direction; subtract if in opposite directions.

• Perpendicular Vectors: Use the Pythagorean theorem to find the resultant.

Vector Components and Decomposition

Breaking Vectors into Components

Any vector in a plane can be decomposed into x (horizontal) and y (vertical) components using trigonometric functions:

where is the magnitude and is the angle from the x-axis.

Example: For m at above the x-axis:

Recomposing Vectors from Components

Given components and , the magnitude and direction are:

Vector Addition by Components

Procedure

1. Resolve each vector into x and y components.

2. Add all x components to get ; add all y components to get .

3. Find the magnitude and direction of the resultant vector:

Unit Vectors

Definition and Notation

Unit vectors are vectors of length 1 that indicate direction along coordinate axes:

• : x-direction

• : y-direction

• : z-direction

Any vector can be written as:

Example:

Dot Product (Scalar Product)

Definition and Calculation

The dot product of two vectors produces a scalar (number):

Alternatively, using components:

• If vectors are perpendicular, dot product is zero.

• If vectors are parallel, dot product is maximized.

Cross Product (Vector Product)

Definition and Calculation

The cross product of two vectors produces a new vector perpendicular to both:

The direction is determined by the right-hand rule:

• Point fingers along , curl toward ; thumb points in direction of .

In component form:

Example: If and , then:

Directions and Angles in Vector Problems

Describing Directions

• Angles measured counterclockwise from the positive x-axis are positive.

• Angles described as "north of east" or similar require careful drawing and interpretation.

Example: A vector 6 m at 30° north of east has components:

Summary Table: Scalar vs. Vector Quantities

Visualizing Vectors in 3D

Vectors in three dimensions are represented with x, y, and z components. The cross product results in a vector perpendicular to the plane formed by the original vectors.

Key Formulas

• Magnitude from components:

• Direction from components:

• Dot product: or

• Cross product: