Vectors: Definitions, Properties, and Operations

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., $\vec{v}$).

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 $\vec{F}$ exerted by the earth on a satellite is a vector directed toward the center of the earth. Its magnitude is proportional to $1/r^2$, where $r$ 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 $\overrightarrow{PQ}$.

• 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 $\vec{v}$ is denoted by $|\vec{v}|$.

• 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 $\vec{u}$ and $\vec{v}$, place the tail of $\vec{v}$ at the head of $\vec{u}$.

• The sum $\vec{u} + \vec{v}$ is the vector from the tail of $\vec{u}$ to the head of $\vec{v}$.

• Commutativity: $\vec{u} + \vec{v} = \vec{v} + \vec{u}$

Scalar Multiplication

• If $c$ is a real number (scalar) and $\vec{v}$ is a vector, then $c\vec{v}$ is a vector whose magnitude is $|c|$ times that of $\vec{v}$ and points in the same direction if $c > 0$, or the opposite direction if $c < 0$.

• Example: $2\vec{v}$ is twice as long as $\vec{v}$; $-1\vec{v}$ points in the opposite direction.

Properties of Scalar Multiplication

• For scalars $c$ and $d$, and vector $\vec{v}$: $(c + d)\vec{v} = c\vec{v} + d\vec{v}$

• For scalar $c$ and vectors $\vec{u}$ and $\vec{v}$: $c(\vec{u} + \vec{v}) = c\vec{u} + c\vec{v}$

Zero Vector

• The zero vector $\vec{0}$ has zero length and no direction.

• Properties:

  • For any vector $\vec{v}$, $\vec{v} + \vec{0} = \vec{v}$

  • For any vector $\vec{v}$, $0\vec{v} = \vec{0}$

• The zero vector is considered parallel to every vector.

Inverse and Subtraction of Vectors

• The additive inverse of $\vec{v}$ is $-\vec{v}$, such that $\vec{v} + (-\vec{v}) = \vec{0}$.

• Vector subtraction is defined as $\vec{u} - \vec{v} = \vec{u} + (-\vec{v})$.

Unit Vectors

• A unit vector has length 1.

• To obtain a unit vector in the direction of $\vec{v}$ (where $\vec{v} \neq \vec{0}$):

  • $\vec{u} = \dfrac{\vec{v}}{|\vec{v}|}$

  • $\vec{u}$ has the same direction as $\vec{v}$ and length 1.

Position Vectors and Coordinate Representation

Position Vectors

• Given a coordinate plane, a vector $\vec{v}$ can be represented by placing its tail at the origin and its head at $(v_1, v_2)$.

• The position vector is $\vec{v} = \langle v_1, v_2 \rangle$.

• Two vectors $\vec{u} = \langle u_1, u_2 \rangle$ and $\vec{v} = \langle v_1, v_2 \rangle$ are equal if and only if $u_1 = v_1$ and $u_2 = v_2$.

Example: Coordinate Representation

• Example: $\vec{w} = \langle 3, -5 \rangle$ is a vector from the origin to the point (3, -5) in the plane.

Vector Operations in Coordinates

• Let $\vec{u} = \langle u_1, u_2 \rangle$ and $\vec{v} = \langle v_1, v_2 \rangle$.

• Addition: $\vec{u} + \vec{v} = \langle u_1 + v_1, u_2 + v_2 \rangle$

• Subtraction: $\vec{u} - \vec{v} = \langle u_1 - v_1, u_2 - v_2 \rangle$

• Scalar Multiplication: $c\vec{u} = \langle c u_1, c u_2 \rangle$

Length (Magnitude) of a Vector

Formula for Magnitude

• The length of $\vec{v} = \langle v_1, v_2 \rangle$ is given by:

  $|\vec{v}| = \sqrt{v_1^2 + v_2^2}$

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

Example Calculations

• Let $\vec{u} = 3\hat{i} + 4\hat{j}$ and $\vec{v} = 2\hat{i} - 5\hat{j}$.

• (a) $2\vec{u} - 3\vec{v}$: Compute by multiplying and subtracting the components.

• (b) $|\vec{u}| = \sqrt{3^2 + 4^2} = 5$

• (c) To find a vector of length 3 in the direction of $\vec{v}$, first find the unit vector in the direction of $\vec{v}$, 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.