PGDDSA Study · Semester 1

Core Titles

Key headlines and terms for quick recall

Dot (scalar) product $\vec{a} \cdot \vec{b} = \|\vec{a}\|\|\vec{b}\| \cos\theta$
Component form $\vec{a}\cdot\vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$
Cross (vector) product $\vec{a} \times \vec{b}$
Magnitude $\|\vec{a} \times \vec{b}\| = \|\vec{a}\| \|\vec{b}\| \sin\theta$
Direction: right-hand rule, perpendicular to both
Triple product $\vec{a} \cdot (\vec{b} \times \vec{c})$ — volume of parallelepiped

Basic Idea

What it is, why it matters, how it works

Dot product (scalar)

$\vec{a} \cdot \vec{b} = \|\vec{a}\| \|\vec{b}\| \cos\theta = a_1 b_1 + a_2 b_2 + a_3 b_3$

Returns a scalar. Measures how much $\vec{a}$ and $\vec{b}$ point in the same direction.

Key results.

$\vec{a} \cdot \vec{a} = \|\vec{a}\|^2$
$\vec{a} \perp \vec{b} \iff \vec{a} \cdot \vec{b} = 0$
Angle: $\cos\theta = \dfrac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \|\vec{b}\|}$
Work done by force $\vec{F}$ over displacement $\vec{d}$ is $\vec{F} \cdot \vec{d}$ .

Cross product (vector)

$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

Returns a vector perpendicular to both $\vec{a}$ and $\vec{b}$ , with magnitude $\|\vec{a} \times \vec{b}\| = \|\vec{a}\| \|\vec{b}\| \sin\theta = \text{area of parallelogram}$ and direction by the right-hand rule.

Key results.

$\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$ (anti-commutative)
$\vec{a} \times \vec{a} = \vec{0}$
$\vec{a} \parallel \vec{b} \iff \vec{a} \times \vec{b} = \vec{0}$
Useful for normals to planes, torque ( $\vec{r} \times \vec{F}$ ), angular momentum.

Scalar triple product

$[\vec{a}, \vec{b}, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})$ Equals the signed volume of the parallelepiped with edges $\vec{a}, \vec{b}, \vec{c}$ . Zero iff the three vectors are coplanar.

Why this matters in Data Science

Dot product is used in cosine similarity, neural-network forward passes, kernels. Cross product appears in 3D geometry — important for graphics, robotics and physics-based ML.

Mind Map

Visual structure of the concept

DOT vs CROSS
├── DOT  a·b  (scalar)
│   ├── = ‖a‖‖b‖ cos θ
│   ├── = Σ aᵢbᵢ
│   ├── a·a = ‖a‖²
│   └── a⊥b ⇔ a·b = 0
├── CROSS  a×b  (vector)
│   ├── Magnitude: ‖a‖‖b‖ sin θ
│   ├── Direction: right-hand rule
│   ├── Perpendicular to both
│   ├── Anti-commutative
│   └── ‖a×b‖ = area of parallelogram
└── Triple product  a·(b×c)
    └── Volume of parallelepiped

Exam Q&A

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

Part A (2 marks each)

Q1. State the dot product formula. $\vec{a} \cdot \vec{b} = \|\vec{a}\| \|\vec{b}\| \cos\theta = a_1 b_1 + a_2 b_2 + a_3 b_3$ .

Q2. State the magnitude of the cross product. $\|\vec{a} \times \vec{b}\| = \|\vec{a}\| \|\vec{b}\| \sin\theta$ , equal to the area of the parallelogram formed by the two vectors.

Q3. Find the angle between $\vec{a} = \langle 1, 0, 1 \rangle$ and $\vec{b} = \langle 0, 1, 1 \rangle$ . $\vec{a} \cdot \vec{b} = 1$ , $\|\vec{a}\| = \|\vec{b}\| = \sqrt{2}$ , $\cos\theta = 1/2 \Rightarrow \theta = 60°$ .

Q4. What does $\vec{a} \cdot (\vec{b} \times \vec{c}) = 0$ imply? The three vectors are coplanar.

Part B (20 marks)

Q. Define dot and cross product of vectors. Derive expressions in component form and give geometrical meaning. Given $\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$ , find $\vec{a} \cdot \vec{b}$ , $\vec{a} \times \vec{b}$ , the angle between them, and the area of the parallelogram they span.

Dot product. $\vec{a} \cdot \vec{b} = \|\vec{a}\| \|\vec{b}\| \cos\theta$ Component form. Write $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ etc. Using $\hat{i}\cdot\hat{i} = 1$ , $\hat{i}\cdot\hat{j} = 0$ : $\vec{a}\cdot\vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3.$ Geometric meaning. Scalar projection times magnitude — measures alignment. Zero ⇒ perpendicular.

Cross product. $\vec{a} \times \vec{b} = \|\vec{a}\| \|\vec{b}\| \sin\theta \;\hat{n}$ where $\hat{n}$ is the unit normal by the right-hand rule. Component form: $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}.$ Geometric meaning. The magnitude is the area of the parallelogram with sides $\vec{a}$ and $\vec{b}$ ; the direction is perpendicular to both.

Computations.

Dot product: $\vec{a} \cdot \vec{b} = (1)(2) + (2)(1) + (1)(-1) = 2 + 2 - 1 = 3$ .

Cross product: $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 1 \\ 2 & 1 & -1 \end{vmatrix} = \hat{i}(2 \cdot -1 - 1 \cdot 1) - \hat{j}(1 \cdot -1 - 1 \cdot 2) + \hat{k}(1 \cdot 1 - 2 \cdot 2) = -3\hat{i} + 3\hat{j} - 3\hat{k}.$

Angle. $\|\vec{a}\| = \sqrt{6}, \|\vec{b}\| = \sqrt{6}$ . $\cos\theta = \dfrac{3}{6} = \dfrac{1}{2} \Rightarrow \theta = 60°$ .

Area of parallelogram. $\|\vec{a} \times \vec{b}\| = \sqrt{9 + 9 + 9} = 3\sqrt{3}$ .

Dot and Cross Product of Two Vectors