PGDDSA Study · Semester 1

Core Titles

Key headlines and terms for quick recall

Vector — magnitude + direction
Position vector $\vec{OP}$
Components $\vec{v} = \langle a, b, c \rangle$ or $a\hat{i} + b\hat{j} + c\hat{k}$
Magnitude $\|\vec{v}\| = \sqrt{a^2 + b^2 + c^2}$
Unit vector $\hat{v} = \vec{v} / \|\vec{v}\|$
Standard basis $\hat{i}, \hat{j}, \hat{k}$
Vector addition, Scalar multiplication

Basic Idea

What it is, why it matters, how it works

Scalars vs vectors

A scalar has only magnitude (e.g., temperature, mass). A vector has both magnitude and direction (e.g., velocity, force).

Representing a vector in 3D

A vector $\vec{v}$ in $\mathbb{R}^3$ is written: $\vec{v} = \langle a, b, c \rangle = a \hat{i} + b \hat{j} + c \hat{k}$ where $\hat{i}, \hat{j}, \hat{k}$ are the standard unit vectors along the $x, y, z$ axes.

The position vector of a point $P(x, y, z)$ is $\vec{OP} = \langle x, y, z \rangle$ .

Magnitude

$\|\vec{v}\| = \sqrt{a^2 + b^2 + c^2}$

Unit vector

A vector of magnitude 1 in the same direction: $\hat{v} = \frac{\vec{v}}{\|\vec{v}\|}$

Operations

Addition: $\vec{u} + \vec{v} = \langle u_1 + v_1, u_2 + v_2, u_3 + v_3 \rangle$ (component-wise; parallelogram law geometrically).
Scalar multiplication: $k \vec{v} = \langle k v_1, k v_2, k v_3 \rangle$ — scales magnitude, reverses direction if $k < 0$ .

Vector from one point to another

$\vec{AB} = \vec{OB} - \vec{OA} = \langle b_1 - a_1, b_2 - a_2, b_3 - a_3 \rangle$ .

Why this matters in Data Science

A datapoint with $n$ features is a vector in $\mathbb{R}^n$ . All ML operations are vector operations: distance, dot product, projection.

Mind Map

Visual structure of the concept

VECTORS IN SPACE
├── Definition: magnitude + direction
├── Components ⟨a, b, c⟩  =  aî + bĵ + ck̂
├── Magnitude ‖v‖ = √(a² + b² + c²)
├── Unit vector v̂ = v / ‖v‖
├── Standard basis î, ĵ, k̂
└── Operations
    ├── Addition  (component-wise)
    ├── Scalar mult.  kv
    └── Vector from A to B  =  ⟨b₁−a₁, b₂−a₂, b₃−a₃⟩

Exam Q&A

Part A (2 marks) and Part B (20 marks) style questions

Part A (2 marks each)

Q1. Define a position vector. The vector from the origin $O$ to the point $P$ , denoted $\vec{OP}$ .

Q2. Find a unit vector in the direction of $\vec{v} = \langle 1, 2, 2 \rangle$ . $\|\vec{v}\| = \sqrt{1+4+4} = 3$ , so $\hat{v} = \langle 1/3, 2/3, 2/3 \rangle$ .

Q3. If $A(1,2,3)$ and $B(4,6,3)$ , find $\vec{AB}$ and its magnitude. $\vec{AB} = \langle 3, 4, 0 \rangle$ , $\|\vec{AB}\| = \sqrt{9 + 16} = 5$ .

Q4. State the parallelogram law of vector addition. If two vectors are represented by adjacent sides of a parallelogram from a common point, their sum is the diagonal of the parallelogram.

Part B (20 marks)

Q. Define vectors in space. Derive magnitude and unit vector. Given $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{b} = -\hat{i} + 4\hat{j} - 2\hat{k}$ , find (i) $\vec{a} + \vec{b}$ , (ii) $|\vec{a} - \vec{b}|$ , (iii) the unit vector along $2\vec{a} - \vec{b}$ .

Definitions. A vector in $\mathbb{R}^3$ has both magnitude and direction. Written as $\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}$ .

Magnitude: $\|\vec{v}\| = \sqrt{a^2 + b^2 + c^2}$ — Pythagoras in 3D.

Unit vector: $\hat{v} = \dfrac{\vec{v}}{\|\vec{v}\|}$ . Always has magnitude 1.

Computations.

(i) Sum. $\vec{a} + \vec{b} = (2-1)\hat{i} + (-3+4)\hat{j} + (1-2)\hat{k} = \hat{i} + \hat{j} - \hat{k}$ .

(ii) Difference. $\vec{a} - \vec{b} = 3\hat{i} - 7\hat{j} + 3\hat{k}$ . Magnitude: $\sqrt{9 + 49 + 9} = \sqrt{67}$ .

(iii) Unit vector along $2\vec{a} - \vec{b}$ . $2\vec{a} = 4\hat{i} - 6\hat{j} + 2\hat{k}$ . $2\vec{a} - \vec{b} = 5\hat{i} - 10\hat{j} + 4\hat{k}$ . Magnitude: $\sqrt{25 + 100 + 16} = \sqrt{141}$ . Unit vector: $\dfrac{1}{\sqrt{141}}(5\hat{i} - 10\hat{j} + 4\hat{k})$ .

Vectors in Space