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Calc-3 Discussion 02

Discussion Topics covered: Dot product, orthogonality, scalar and vector projections, cross product, geometric determinants, 3D line representations, point-to-line distance, line relationship classifications, single-variable derivative rules, finding domains of algebraic functions

Part A: Review This Week's Lecture Topics (~30 mins)

The Dot Product & Vector Projections

  • Computing algebraic dot products (u⃗⋅v⃗=u1v1+u2v2+u3v3\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 + u_3 v_3) and geometric dot products (u⃗⋅v⃗=∥u⃗∥∥v⃗∥cos⁡θ\vec{u} \cdot \vec{v} = \|\vec{u}\|\|\vec{v}\|\cos\theta) to quantify directional alignment and enforce the scalar output rule
  • Applying the orthogonality condition (u⃗⋅v⃗=0\vec{u} \cdot \vec{v} = 0) and evaluating the exact angle between 3D vectors via cos⁡θ=u⃗⋅v⃗∥u⃗∥∥v⃗∥\cos\theta = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\| \|\vec{v}\|}
  • Decomposing vectors onto structural axes using scalar projections compv⃗u⃗=u⃗⋅v⃗∥v⃗∥\text{comp}_{\vec{v}}\vec{u} = \frac{\vec{u} \cdot \vec{v}}{\|\vec{v}\|} and vector projections projv⃗u⃗=(u⃗⋅v⃗∥v⃗∥2)v⃗\text{proj}_{\vec{v}}\vec{u} = \left(\frac{\vec{u} \cdot \vec{v}}{\|\vec{v}\|^2}\right)\vec{v}
  • Modeling mechanical work (W=F⃗⋅d⃗W = \vec{F} \cdot \vec{d}) along constant force vectors and decomposing axial tension/compression in bridge truss members

The Cross Product & Spatial Geometry

  • Evaluating 3×33 \times 3 matrix determinants to compute vector cross products u⃗×v⃗\vec{u} \times \vec{v}, applying the right-hand rule, and accounting for anti-commutativity (u⃗×v⃗=−(v⃗×u⃗)\vec{u} \times \vec{v} = -(\vec{v} \times \vec{u}))
  • Constructing unit vectors orthogonal to a given plane or pair of non-parallel vectors: n^=u⃗×v⃗∥u⃗×v⃗∥\hat{n} = \frac{\vec{u} \times \vec{v}}{\|\vec{u} \times \vec{v}\|}
  • Calculating geometric areas of parallelograms (∥u⃗×v⃗∥\|\vec{u} \times \vec{v}\|) and 3D triangles (12∥u⃗×v⃗∥\frac{1}{2}\|\vec{u} \times \vec{v}\|) spanned by displacement vectors
  • Modeling physical torque moments (τ⃗=r⃗×F⃗\vec{\tau} = \vec{r} \times \vec{F}) on mechanical linkages and calculating 3D parallelepiped volumes via the scalar triple product ∣u⃗⋅(v⃗×w⃗)∣|\vec{u} \cdot (\vec{v} \times \vec{w})|

Equations of Lines in 3D Space

  • Constructing vector equations r⃗(t)=r⃗0+tv⃗\vec{r}(t) = \vec{r}_0 + t\vec{v}, parametric equations (x=x0+at,y=y0+bt,z=z0+ctx = x_0 + at, y = y_0 + bt, z = z_0 + ct), and symmetric equations (x−x0a=y−y0b=z−z0c\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}) from points and direction vectors
  • Extracting direction vectors v⃗=⟨a,b,c⟩\vec{v} = \langle a, b, c \rangle from two points, parallel trajectories, or mutual orthogonality constraints
  • Handling zero-component directions (b=0  ⟹  y=y0b = 0 \implies y = y_0) and restricting parameter domains for bounded line segments (0≤t≤10 \le t \le 1)

Distances and Spatial Line Relationships

  • Calculating the shortest perpendicular clearance distance from an external point to a line in space via the cross-product formula d=∥PM⃗×v⃗∥∥v⃗∥d = \frac{\|\vec{PM} \times \vec{v}\|}{\|\vec{v}\|}
  • Classifying pairs of lines in R3\mathbb{R}^3 as identical, parallel, intersecting, or skew by testing direction vector proportionality (u⃗=kv⃗\vec{u} = k\vec{v}) and setting up independent two-parameter linear systems (tt and ss)

Part B: Next Week's Prerequisite Refresher (~30 mins)

Review the following prerequisite skills:

Part C: General Q/A

Open the floor to questions from students regarding any topic