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Vector Algebra
Introduction
Some Basic Concepts
Types of Vectors
Practice: Vectors and Their Types
Addition of Vectors
Multiplication of a Vector by a Scalar
Practice: Components, Sums and Section Formula
Product of Two Vectors
Practice: Dot Product and Projections
Practice: Cross Product
Miscellaneous Examples
Chapter Review: Mixed Practice
Summary
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Vector AlgebraIntroduction

Reading time: ~10 min

It is Uttarayan, and the sky over the old city is full of kites. Aman's string snaps, and his kite drifts down into the open ground behind his house. From the next terrace Riya saw exactly where it landed. "It is 5 metres from where you are standing!" she shouts.

Aman runs down to the ground. Is that enough to find the kite?

You are Aman. The grid is the ground, one square to a metre, and Aman stands at the centre. Tap grid crossings that fit Riya's first clue; then use her second clue to find the kite.

"5 metres" told Aman how far, but not which way, so a whole circle of places fitted it. Riya's second clue added a direction, and only one place was left: 5 m towards the tank.

Many questions in daily life are like Riya's first clue. How tall are you? 1.6 metres. How hot is it today? 32 °C. One real number — a magnitude — answers them completely. Such quantities are called scalars: length, mass, time, distance, speed, area, volume, temperature, work, money, voltage, density and resistance are all scalars.

Other questions need a direction as well. Where did the kite land? How should a footballer kick the ball to pass it to a teammate? The kick needs a strength and a direction — towards the teammate. Quantities with both a magnitude and a direction are called vectors: displacement, velocity, acceleration, force, weight, momentum and electric field intensity are examples. In this chapter you will learn to add vectors, stretch them, take them apart along the axes, and multiply them in two very different ways.

Carried From Here to There

The word vector comes from the Latin vectus, "carried": a vector carries you from one point to another. The idea is much older than the word.

Drag these milestones into the order in which they happened, earliest at the top.

Caspar Wessel and Jean-Robert Argand draw complex numbers as arrows in a plane (around 1800)
In Aristotle's time, two forces are combined with a parallelogram rule (4th century BCE)
Josiah Willard Gibbs and Oliver Heaviside create modern vector analysis (1880s)
Simon Stevin studies how forces add, in Principles of the Art of Weighing (1586)
William Rowan Hamilton first calls a directed line segment a "vector" (1853)

More than two thousand years separate the first step from the last. Aristotle's parallelogram rule for forces, Stevin's study of how forces add, and the arrows of Wessel and Argand all fed into Hamilton's quaternions — four-part numbers a + b\hat{i} + c\hat{j} + d\hat{k} that he invented to multiply directions in space. Gibbs and Heaviside kept the useful "vector part" and built the toolkit that physicists and engineers use every day — the toolkit of this chapter.

Sina