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Feynman’s Lectures on Physics: Chapter 11-6 – Newton’s Laws in Vector Notation

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From three equations to one universal law One of the great achievements of Newtonian mechanics was to establish a mathematical framework capable of describing the motion of objects, from falling stones to orbiting planets. Yet the way we express these laws mathematically has an important bearing on how we understand their universality. In Chapter 11-6 of The Feynman Lectures on Physics , Richard Feynman brings together the ideas developed in the preceding sections on vectors, vector addition and rotations to show how Newton’s laws can be expressed in a form that is independent of our choice of coordinate system. The central idea is deceptively simple: Newton’s laws are not fundamentally statements about motion along the x, y and z axes. They are statements about physical quantities in space. By expressing them using vectors, we can write a single law that applies equally well regardless of how we orient our coordinate system. Acceleration as a vector We are accustomed to thinking of...
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Feynman's Lectures on Physics – Chapter 11.5: Vector Algebra

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From the geometry of arrows to the mathematics of motion, Richard Feynman explores how vectors can be combined, transformed and differentiated while retaining their essential character. Introduction: More than three numbers There is something deceptively simple about a vector. We draw an arrow, give it a length and a direction, and use it to represent quantities such as displacement, velocity and force. Yet beneath this familiar geometrical picture lies a rather more interesting mathematical idea. In the preceding sections of Chapter 11, Feynman has established that a vector is not simply a collection of three numbers. It is a quantity whose components change in a particular, mathematically prescribed way when we rotate our coordinate system. This distinction is fundamental. It is what separates a genuine physical vector from an arbitrary collection of numerical values. In this section, Feynman begins to develop the algebra of vectors. How can we add them? What does it mean to multi...
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