Vector Operations

Source lecture(s): PC604 Lec1-2, PHY621 Lec1

Intuition

Vectors carry magnitude and direction. Every oriented physical quantity—force, velocity, torque—needs vector algebra.

Formal Definition

A vector $\vec{A}\in\mathbb{R}^3$ has components $A_x,A_y,A_z$. Dot product yields a scalar; cross product yields a vector perpendicular to both.

Mathematical Formulation

\[\vec{A}\cdot\vec{B}=A_xB_x+A_yB_y+A_zB_z$$ $$\vec{A}\times\vec{B}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ A_x&A_y&A_z\\ B_x&B_y&B_z\end{vmatrix}\]

Derivation

The dot product follows from projecting one vector onto the other. The cross product magnitude equals the area of the parallelogram formed by $\vec{A}$ and $\vec{B}$.

Worked Example

For $\vec{A}=3\hat{i}+4\hat{j}$ and $\vec{B}=\hat{i}+2\hat{j}+2\hat{k}$, $\vec{A}\cdot\vec{B}=11$ and $|\vec{A}|=5$.

Common Mistakes

Using $|\vec{A}||\vec{B}|$ for non-parallel vectors in the dot product.
Forgetting anti-commutativity $\vec{A}\times\vec{B}=-\vec{B}\times\vec{A}$.

Quiz

Q1. When is $\vec{A}\times\vec{B}$ parallel to $\vec{A}$?

Answer

When $\vec{B}$ has no component perpendicular to $\vec{A}$.

Q2. What does $\vec{A}\cdot\vec{B}=0$ imply?