Ep 32: The Product That Only Exists in Three Dimensions | The Cross Product

Last episode we built the stage --- three perpendicular axes, the distance formula, the equation of a sphere, surfaces and their level curves. Today we install the machinery for \emph{moving around} on that stage. A \emph{vector} is an arrow with a length and a direction, or equivalently a tuple of numbers \langle v_1, v_2, v_3 \rangle. Vectors add tip-to-tail, scale by stretching, have a magnitude |v| = \sqrt{v_1^2 + v_2^2 + v_3^2}, and combine in two genuinely different ways. The \emph{dot product} multiplies two vectors into a number that secretly encodes the angle between them: u \cdot v = |u||v|\cos\theta. Out of one identity falls perpendicularity, projection, and physical work. The \emph{cross product} multiplies two vectors into a third vector, perpendicular to both, whose length is the area of the parallelogram they span. It only exists in \mathbb{R}^3, it has a handedness, and it is the algebraic ghost of Hamilton's quaternions. By the end of this episode we have the toolkit for lines, planes, projections, torque, and every geometric argument the rest of the series will need. 📚 Continues in: → 'Limits and Continuity in Multiple Variables' --- with vectors and geometry in hand, we revisit the most fundamental analytic notion of calculus, now in \mathbb{R}^n. Approaching a point from infinitely many directions changes everything. 🎓 Best for: Calculus 1, 2, and 3 students, AP Calculus AB / BC, anyone reviewing before an exam, and self-learners who want a single coherent course instead of a pile of disconnected YouTube videos. — QED Realized — Clean explanations of the ideas that take a real lecture to land. 🔔 Subscribe so you don't miss the next one:    / @qedrealized   #Calculus #Math #Mathematics #STEM #Vectors #DotProduct #CrossProduct #MultivariableCalculus #Calculus3 — Tags (for search) — vectors tutorial, vectors in r2 r3, ordered tuples as vectors, geometric vs algebraic vectors, vector addition tip to tail, scalar multiplication of vectors, magnitude of a vector formula, dot product definition, dot product formula, dot product geometric meaning, cosine of angle between vectors, orthogonality test dot product zero, vector projection formula, proj_v u derivation, cross product definition, cross product determinant formula, cross product perpendicular vector, area of parallelogram cross product, right hand rule orientation, i cross j equals k, work as dot product, torque as cross product, normal vector to a plane, hamilton quaternions history, gibbs heaviside vector calculus, vector identities, multivariable calculus prerequisites ⏱ Chapters 0:00 The algebra of arrows 0:55 Two perspectives on the same object 2:03 Componentwise, geometrically tip-to-tail 3:08 Pythagoras, applied to an arrow 4:19 Vectors \to a number 6:01 u \cdot v = 0, and the shadow of u on v 7:33 Geometry of u, v, \theta 8:30 Vectors \to a vector (in \mathbb{R}^3 only) 9:48 Orientation, and the area of a parallelogram 11:11 Curl from u to v, thumb along u \times v 12:01 u = \langle 1, 2, 3\rangle, v = \langle 4, 5, 6\rangle 13:24 Same u, v. The cross product. 15:02 Standard basis, by determinant 16:13 Work, torque, planes; Hamilton, Gibbs, Heaviside 17:57 Recap 19:27 Outro 20:21 Thanks for watching