Scalars and Vectors

Definition of Scalar Quantity

A scalar quantity is any quantity that is completely described by a single real number specifying its magnitude. Scalars do not have direction.

• Examples: Mass, temperature, energy, and time are scalar quantities.

• Mathematical Representation: A scalar is just a real number, often denoted by symbols such as c or k.

Definition of Vector Quantity

A vector quantity is any quantity that has both magnitude and direction. Mathematically, a vector is a directed line segment.

• Examples: Displacement, velocity, acceleration, force, and electric field.

• Notation: Vectors are often denoted by boldface letters (e.g., v) or with an arrow above (e.g., ).

Examples and Applications

• Example: Consider water moving in a stream. In a small region, the water moves at a certain speed (magnitude) and in a certain direction, making its velocity a vector.

• Example: The gravitational force exerted by the earth on a satellite is a vector directed toward the center of the earth. Its magnitude is proportional to , where is the distance from the earth's center to the satellite.

Basic Vector Concepts

Vector Notation and Terminology

• If a vector extends from point P to point Q, it is denoted as .

• Tail (Initial Point): The starting point of the vector.

• Head (Terminal Point): The ending point of the vector.

Magnitude (Length) of a Vector

• The magnitude or length of a vector is denoted by .

• Vectors are equal if they have the same length and direction, regardless of their initial position.

Vector Operations

Vector Addition

Vectors can be added geometrically using the "tip-to-tail" method:

• To add and , place the tail of at the head of .

• The sum is the vector from the tail of to the head of .

• Commutativity:

Scalar Multiplication

• If is a real number (scalar) and is a vector, then is a vector whose magnitude is times that of and points in the same direction if , or the opposite direction if .

• Example: is twice as long as ; points in the opposite direction.

Properties of Scalar Multiplication

• For scalars and , and vector :

• For scalar and vectors and :

Zero Vector

• The zero vector has zero length and no direction.

• Properties:

  • For any vector ,

  • For any vector ,

• The zero vector is considered parallel to every vector.

Inverse and Subtraction of Vectors

• The additive inverse of is , such that .

• Vector subtraction is defined as .

Unit Vectors

• A unit vector has length 1.

• To obtain a unit vector in the direction of (where ):

  • has the same direction as and length 1.

Position Vectors and Coordinate Representation

Position Vectors

• Given a coordinate plane, a vector can be represented by placing its tail at the origin and its head at .

• The position vector is .

• Two vectors and are equal if and only if and .

Example: Coordinate Representation

• Example: is a vector from the origin to the point (3, -5) in the plane.

Vector Operations in Coordinates

• Let and .

• Addition:

• Subtraction:

• Scalar Multiplication:

Length (Magnitude) of a Vector

Formula for Magnitude

• The length of is given by:

• This formula comes from the Pythagorean theorem, treating the vector as the hypotenuse of a right triangle.

Example Calculations

• Let and .

• (a) : Compute by multiplying and subtracting the components.

• (b)

• (c) To find a vector of length 3 in the direction of , first find the unit vector in the direction of , then multiply by 3.

Additional info: These notes provide foundational concepts for vectors, which are essential in multivariable calculus and physics. Understanding vector operations is crucial for topics such as vector-valued functions, gradients, and applications in mechanics.