Lecture-1 Vectors Notes

Vectors: Overview

Vectors are geometrical entities that have both magnitude and direction. They are commonly represented by arrows, where the length of the arrow corresponds to the magnitude and the arrowhead indicates the direction. The initial point and the terminal point define the displacement from one point to another. The use of vectors began in the late 19th century with the development of electromagnetic induction. The word vector is from Latin, meaning carrier, reflecting the idea that a vector carries a point A to a point B. The magnitude of a vector is the length of the segment AB, and the direction is the direction of the displacement from A to B. Vectors have wide applications across mathematics, physics, engineering, and many other fields. In most contexts, vectors are written in bold lowercase letters, such as a or b.

Components of Vectors

A vector quantity possesses two key characteristics: magnitude and direction. When comparing two vectors of the same type, these two quantities are taken into account together. In a two-dimensional coordinate system, any vector can be decomposed into its x-component and y-component. More generally, a vector in n dimensions has components denoted by $a<em>1, a</em>2, \dots, a_n$.

Magnitude of Vectors

The magnitude of a vector with components $a<em>1, a</em>2, \dots, a<em>n$ is given by $|a| = \sqrt{a</em>1^2 + a<em>2^2 + \cdots + a</em>n^2}$
The magnitude is a scalar value.

Types of Vectors

Vectors can be categorized based on magnitude, direction, and their relationship to other vectors.

Zero Vector

A vector with zero magnitude is called the zero vector. It has zero magnitude and no direction and serves as the additive identity. The zero vector is denoted as 0 or as the boldface zero vector $\mathbf{0}$.

Unit Vector

A unit vector has magnitude equal to 1 and is used to denote direction. The magnitude of a unit vector is $|\hat{u}| = 1$.

Equal Vectors

Two or more vectors are equal if their corresponding components are equal. Equal vectors have the same magnitude and the same direction, although their initial and terminal points may differ.

Negative Vector

A vector is the negative of another if they have the same magnitude but opposite directions. If vectors $\mathbf{A}$ and $\mathbf{B}$ have equal magnitudes but opposite directions, then $\mathbf{A} = -\mathbf{B}$.

Parallel Vectors

Two or more vectors are parallel if they have the same direction, though their magnitudes may differ. The angles of their directions differ by zero degrees. Vectors whose direction angles differ by 180 degrees are called antiparallel vectors, i.e., they point in opposite directions.

Orthogonal Vectors

Vectors are orthogonal when the angle between them is $90^{\circ}$. In other words, the dot product of orthogonal vectors is always zero:
$\mathbf{a} \cdot \mathbf{b} = 0 \quad \text{if} \quad \alpha = 90^{\circ}.$

Operations on Vectors

Basic vector operations can be performed without reference to a coordinate system. The primary operations are addition, subtraction, and scalar multiplication, along with two forms of multiplication: the dot product and the cross product. Additionally, the scalar triple product is mentioned as part of the vector operations.

Addition of Vectors

Adding vectors is done component-wise:
$\mathbf{a} + \mathbf{b} = (a<em>1 + b</em>1, a<em>2 + b</em>2, \dots, a<em>n + b</em>n)$
The addition of vectors is commutative and associative:
$\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}$
$(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})$
And the zero vector acts as the additive identity: $\mathbf{a} + \mathbf{0} = \mathbf{a}$

Subtraction of Vectors

Vector subtraction is defined similarly by changing the sign of corresponding components:
$\mathbf{a} - \mathbf{b} = (a<em>1 - b</em>1, a<em>2 - b</em>2, \dots, a<em>n - b</em>n)$

Scalar Multiplication of Vectors

A scalar multiplies each component of a vector:
$\lambda \mathbf{a} = (\lambda a<em>1, \lambda a</em>2, \dots, \lambda a_n)$
If $\lambda$ is negative, the resulting vector reverses direction (rotates by 180 degrees).
The scalar multiplication is distributive over vector addition:
$\lambda (\mathbf{a} + \mathbf{b}) = \lambda \mathbf{a} + \lambda \mathbf{b}$

Scalar Triple Product of Vectors

The scalar triple product is listed as part of the vector operations. It is defined for vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ as
$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$
This yields a scalar value and is related to volumes in geometry.

Multiplication of Vectors

Vectors can be multiplied, but there are different types of multiplication. The two primary forms discussed are the dot product and the cross product. The dot product yields a scalar, while the cross product (not elaborated here) yields a vector perpendicular to both operands.

Dot Product of Vectors

The dot product of two vectors is given component-wise by
$(\mathbf{a}, \mathbf{b}) = a<em>1 b</em>1 + a<em>2 b</em>2 + \cdots + a<em>n b</em>n$
It can also be expressed in terms of magnitudes and the angle between the vectors as
$(\mathbf{a}, \mathbf{b}) = |\mathbf{a}| |\mathbf{b}| \cos \alpha$

Angle Between Two Vectors

Let $\alpha$ be the angle between vectors $\mathbf{a}$ and $\mathbf{b}$. The dot product is related to this angle by
$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \alpha$
Thus the angle can be determined from the dot product as
$\alpha = \arccos\left( \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \right)$
The angle indicates how far apart the directions of the two vectors are.

Orthogonality Revisited

Two vectors are orthogonal if and only if the angle between them is $90^{\circ}$, which is equivalent to having a dot product of zero:
$\mathbf{a} \cdot \mathbf{b} = 0 \quad \text{when} \quad \alpha = 90^{\circ}.$

Cross Product (Note)

The material mentions the cross product as another way to multiply vectors, but it does not provide its definition here. In three dimensions, the cross product yields a vector perpendicular to both operands and has magnitude $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin \theta$, with direction given by the right-hand rule. This topic is typically covered in subsequent sections.

Connections to Foundational Principles and Real-World Relevance

Vectors provide a language for describing displacements, forces, velocities, and fields in physics and engineering. The magnitude–direction pair captures both how much and in what direction a quantity acts. The orthogonality concept underpins projections and decompositions of forces, motions, and signals. The dot product connects to work done by a force along a displacement, as work is the projection of force in the direction of motion, and is given by $W = \mathbf{F} \cdot \Delta \mathbf{r}$ when appropriate. The angle between vectors informs about similarity of directions, correlations between signals, and alignment in mechanical systems. Understanding these operations equips students to solve problems involving rotations, projections, and decompositions in higher dimensions as well.