Three-dimensional space Study Guide

📖 Core Concepts Three‑dimensional space – points need three coordinates \((x,y,z)\); the set of all such triples is \(\mathbb{R}^3\). Coordinate systems – Cartesian \((x,y,z)\); cylindrical \((\rho,\phi,z)\); spherical \((r,\theta,\phi)\). Line & plane determination – 2 points → line; 3 non‑collinear points → unique plane. Skew lines – non‑intersecting and non‑coplanar (only possible in 3‑D). Vectors – ordered triple \([v1,v2,v3]\); magnitude \(\|\mathbf v\|=\sqrt{v1^2+v2^2+v3^2}\). Dot product – scalar \( \mathbf A\!\cdot\!\mathbf B = A1B1+A2B2+A3B3 = \|\mathbf A\|\|\mathbf B\|\cos\theta\). Cross product – vector \(\mathbf A\times\mathbf B\) perpendicular to both; components \((A2B3-A3B2,\;A3B1-A1B3,\;A1B2-A2B1)\); magnitude \(\|\mathbf A\|\|\mathbf B\|\sin\theta\). Gradient, divergence, curl – \(\nabla f\), \(\nabla\!\cdot\!\mathbf F\), \(\nabla\times\mathbf F\) respectively. Fundamental theorems – line integral of a gradient field, Stokes’ theorem, Divergence (Gauss) theorem. Sphere & ball – surface area \(A=4\pi r^{2}\); volume \(V=\frac{4}{3}\pi r^{3}\). Rotations – form the Lie group \(SO(3)\); preserve lengths & angles. --- 📌 Must Remember Dot product rule: \(\mathbf A\!\cdot\!\mathbf B = \|\mathbf A\|\|\mathbf B\|\cos\theta\). Cross product rule: \(\|\mathbf A\times\mathbf B\| = \|\mathbf A\|\|\mathbf B\|\sin\theta\); direction given by right‑hand rule. Plane equation (single linear equation) defines a hyperplane (2‑D) in \(\mathbb{R}^3\). Volume of a ball: \(V=\frac{4}{3}\pi r^{3}\). Surface area of a sphere: \(A=4\pi r^{2}\). Stokes’ theorem: \(\displaystyle \iint{\Sigma} (\nabla\times\mathbf F)\cdot d\mathbf S = \oint{\partial\Sigma} \mathbf F\cdot d\mathbf r\). Divergence theorem: \(\displaystyle \iiint{V}(\nabla\!\cdot\!\mathbf F)\,dV = \iint{\partial V}\mathbf F\cdot\mathbf n\,dS\). Gradient field: \(\mathbf F=\nabla f \;\Rightarrow\; \intC\mathbf F\cdot d\mathbf r = f(\mathbf r{\text{end}})-f(\mathbf r{\text{start}})\). Cross product existence: only in 3 D (and 7 D) for a binary operation returning a vector. --- 🔄 Key Processes Compute a dot product Write vectors component‑wise. Multiply matching components and sum. Compute a cross product Use the determinant or component formula. Apply right‑hand rule for direction. Find the angle between two vectors \(\displaystyle \theta = \arccos\!\frac{\mathbf A\!\cdot\!\mathbf B}{\|\mathbf A\|\|\mathbf B\|}\). Convert coordinates Cartesian → cylindrical: \(\rho=\sqrt{x^{2}+y^{2}},\;\phi=\tan^{-1}(y/x),\;z=z\). Cartesian → spherical: \(r=\sqrt{x^{2}+y^{2}+z^{2}},\;\theta=\cos^{-1}(z/r),\;\phi=\tan^{-1}(y/x)\). Apply Stokes’ theorem Identify surface \(\Sigma\) bounded by curve \(C\). Compute \(\nabla\times\mathbf F\). Evaluate surface integral \(\iint{\Sigma}(\nabla\times\mathbf F)\cdot d\mathbf S\). Apply Divergence theorem Identify closed volume \(V\) with boundary \(\partial V\). Compute \(\nabla\!\cdot\!\mathbf F\). Integrate over \(V\) or compute flux \(\iint{\partial V}\mathbf F\cdot\mathbf n\,dS\). --- 🔍 Key Comparisons Dot vs. Cross Dot: scalar, measures projection, uses \(\cos\theta\). Cross: vector, measures area spanned, uses \(\sin\theta\), right‑hand rule. Cylindrical vs. Spherical coordinates Cylindrical: \((\rho,\phi,z)\) – keep \(z\) as in Cartesian, good for problems with axial symmetry. Spherical: \((r,\theta,\phi)\) – radial distance from origin, ideal for central‑force or radial symmetry. Intersecting vs. Parallel vs. Skew lines Intersecting: share a point. Parallel: never meet, lie in same plane. Skew: never meet and not coplanar. Gradient vs. Divergence vs. Curl Gradient: points in direction of greatest increase of a scalar field. Divergence: scalar describing net “source/sink” strength of a vector field. Curl: vector describing local rotation (circulation) of a vector field. --- ⚠️ Common Misunderstandings Cross product in 2‑D – impossible; you must embed vectors in \(\mathbb{R}^3\) (the result points out of the plane). Parallel vs. Skew – parallel lines are always coplanar; skew lines are not parallel. Surface vs. Volume integrals – mixing up \(dS\) (area element) with \(dV\) (volume element) leads to wrong units. Stokes vs. Divergence – Stokes relates a surface integral of curl to a line integral; Divergence relates a volume integral of divergence to a flux through the closed surface. Right‑hand rule sign – forgetting the orientation of the normal vector flips the sign of curl‑related results. --- 🧠 Mental Models / Intuition Cross product = “area vector” of the parallelogram spanned by two vectors; its direction is the axis about which the two vectors rotate into each other. Divergence = “how much a field is blowing out of a tiny balloon” placed at a point. Curl = “how much a tiny paddle wheel would spin” if placed in the field. Gradient = “steepest ascent” direction; level surfaces are orthogonal to \(\nabla f\). Skew lines → think of two non‑parallel, non‑intersecting sticks in space; they cannot be brought together without moving out of their planes. --- 🚩 Exceptions & Edge Cases Cross product existence – only defined as a binary vector‑valued operation in \(\mathbb{R}^3\) (and \(\mathbb{R}^7\)). Ruled surfaces – hyperboloid of one sheet and hyperbolic paraboloid consist of straight line families; not every quadric is ruled. Hyperplane – in 3‑D a hyperplane is a plane (2‑D); the term “hyperplane” is more general for higher dimensions. Rotations – not commutative; order matters (e.g., rotate about \(x\) then \(y\) ≠ rotate about \(y\) then \(x\)). --- 📍 When to Use Which Dot product → find angle, projection, work \(W=\mathbf F\!\cdot\!\mathbf d\). Cross product → compute torque, area of parallelogram, normal vector to a plane. Cylindrical coordinates → problems with symmetry around a fixed axis (e.g., infinite cylinder). Spherical coordinates → problems with radial symmetry (e.g., point charge, gravitational field). Stokes’ theorem → when a line integral around a closed curve is easier than a surface integral of curl, or vice‑versa. Divergence theorem → when evaluating flux through a closed surface; often easier to convert to a volume integral. Gradient field test → if \(\nabla\times\mathbf F = \mathbf 0\) on a simply‑connected domain, \(\mathbf F\) is conservative → use fundamental theorem for line integrals. --- 👀 Patterns to Recognize Presence of \(\sin\theta\) → indicates a cross‑product‑type quantity (area, torque). Zero curl → suggests a conservative (gradient) field → path‑independent line integrals. Zero divergence → suggests a solenoidal field → flux through any closed surface is zero. Quadratic equation with all same sign coefficients → ellipsoid; mixed signs → hyperboloid or cone. Two families of straight lines on a surface → ruled surface (hyperboloid‑1‑sheet or hyperbolic paraboloid). --- 🗂️ Exam Traps Mixing up volume and surface formulas – using \(V=\frac{4}{3}\pi r^{3}\) for surface area or vice‑versa. Choosing the wrong coordinate system – applying cylindrical Jacobian \( \rho \, d\rho\, d\phi\, dz\) to a problem better suited to spherical coordinates. Sign error in Stokes’ theorem – orientation of \(\partial\Sigma\) must follow the right‑hand rule relative to the chosen normal. Assuming any two lines are either intersecting or parallel – forgetting the possibility of skew lines. Treating cross product as commutative – \(\mathbf A\times\mathbf B = -(\mathbf B\times\mathbf A)\); forgetting the minus sign flips direction. Using dot product for torque – torque requires cross product \(\boldsymbol{\tau} = \mathbf r \times \mathbf F\). ---