Section 2.2 - Vectors

Section 2.2 - Vectors

Introduction to Vectors

Vectors are fundamental mathematical objects used to represent quantities that have both magnitude and direction. They are widely used in physics, engineering, and mathematics to describe displacement, velocity, force, and other directional quantities.

• Definition: A vector is a quantity with both magnitude and direction, often represented by an arrow or a boldface letter (e.g., v or a).

• Notation: Vectors are typically denoted as \( \vec{v} \) or \( \mathbf{v} \).

• Initial Point and Terminal Point: The initial point (tail) is where the vector starts, and the terminal point (head) is where it ends.

• Direction: The direction of a vector is indicated by the arrowhead, and its length represents the magnitude.

Vector Representation and Notation

Vectors can be represented as directed line segments from one point to another. The vector from point A to point B is written as \( \overrightarrow{AB} \).

• Magnitude: The length of the arrow represents the magnitude of the vector.

• Direction: The arrow points in the direction of the vector.

• Zero Vector: The only vector with no specific direction is the zero vector, denoted by \( \vec{0} \).

Combining Vectors

Vectors can be combined using addition and subtraction. The most common method for vector addition is the Triangle Law.

• Vector Addition: If a and b are vectors positioned so that the initial point of b is at the terminal point of a, then the sum \( \vec{a} + \vec{b} \) is the vector from the initial point of a to the terminal point of b.

• Triangle Law: This method of addition is also called the Triangle Law.

Example 1: Draw the sum of the vectors a and b as shown in the diagram.

Example 2: Draw the difference of the vectors a and b as shown in the diagram.

Example 3: Draw the vector \( \vec{a} - 2\vec{b} \) as shown in the diagram.

Components of Vectors

Vectors can be described in terms of their components in a coordinate system. If the initial point of a vector is at the origin, its terminal point has coordinates \( (n_1, n_2, ..., n_k) \) depending on the dimension.

• 2D and 3D Vectors: In two or three dimensions, a vector \( \vec{v} \) is written as \( (v_1, v_2) \) or \( (v_1, v_2, v_3) \).

• n-Dimensional Vectors: In general, an n-dimensional vector is an ordered n-tuple: \( \vec{v} = (v_1, v_2, ..., v_n) \).

• Vector Space: The set of all n-dimensional vectors is denoted \( \mathbb{R}^n \).

Example 4a: Draw the vector \( \overrightarrow{OP} = (2,5) \) and \( \overrightarrow{OQ} = (0,1,-1) \).

Length (Magnitude) of a Vector

The length or magnitude of a vector \( \vec{v} = (v_1, v_2, v_3) \) is given by:

$| \vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$

• If \( |\vec{v}| = 1 \), the vector is called a unit vector.

• For any nonzero vector \( \vec{v} \), the unit vector in the same direction is \( \frac{\vec{v}}{|\vec{v}|} \).

Vector Operations in Component Form

• Addition: If \( \vec{a} = (a_1, a_2) \) and \( \vec{b} = (b_1, b_2) \), then \( \vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2) \).

• Subtraction: \( \vec{a} - \vec{b} = (a_1 - b_1, a_2 - b_2) \).

• Scalar Multiplication: \( c\vec{a} = (c a_1, c a_2) \) for scalar c.

Example 5: If \( \vec{a} = (4,0,3) \) and \( \vec{b} = (-2,1,5) \), find \( |\vec{a}| \) and the vectors \( 2\vec{a} + \vec{b} \) and \( \vec{a} - 2\vec{b} \).

Properties of Vector Operations

The following properties hold for vectors \( \vec{a}, \vec{b}, \vec{c} \) and scalars c, d:

Property	Equation
Commutativity of Addition	\( \vec{a} + \vec{b} = \vec{b} + \vec{a} \)
Associativity of Addition	\( (\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c}) \)
Distributivity of Scalar Multiplication	\( c(\vec{a} + \vec{b}) = c\vec{a} + c\vec{b} \)
Distributivity over Scalars	\( (c + d)\vec{a} = c\vec{a} + d\vec{a} \)
Associativity of Scalar Multiplication	\( c(d\vec{a}) = (cd)\vec{a} \)
Identity Element	\( 1\vec{a} = \vec{a} \)

Standard Basis Vectors

In three-dimensional space, the standard basis vectors are:

• \( \vec{i} = (1, 0, 0) \)

• \( \vec{j} = (0, 1, 0) \)

• \( \vec{k} = (0, 0, 1) \)

Any vector \( \vec{v} = (v_1, v_2, v_3) \) can be written as a linear combination of these basis vectors:

$\vec{v} = v_1 \vec{i} + v_2 \vec{j} + v_3 \vec{k}$

Example 6: If \( \vec{a} = (2, -1, 5) \) and \( \vec{b} = (-1, 0, 3) \), express the vector \( 2\vec{a} - \vec{b} \) in terms of \( \vec{i}, \vec{j}, \vec{k} \).

Additional info: The notes also reference the geometric interpretation of vectors, the use of coordinates in higher dimensions, and the importance of basis vectors in expressing any vector in space. These concepts are foundational for further study in calculus and linear algebra.