Comprehensive Study Notes on Vector Cross Products in Three-Dimensional Space

Course Logistics and WebAssign Homework

• Homework Assignments and Pace:
  • Homework is assigned via WebAssign.
  • Each section averages around $5$ problems.
  • The standard course pace covers $2$ sections per week.
  • Students experiencing issues with WebAssign registration should seek assistance immediately.

Fundamentals of Vector Operations in 2D and 3D

• Overview of Basic Vector Operations:

  • Vector Addition: Adds two vectors component-wise to yield a new vector ($\mathbf{u} + \mathbf{v}$). Works in both $\mathbb{R}^2$ and $\mathbb{R}^3$.
  • Vector Subtraction: Subtracts one vector from another component-wise to yield a new vector ($\mathbf{u} - \mathbf{v}$). Works in both $\mathbb{R}^2$ and $\mathbb{R}^3$.
  • Scalar Multiplication: Multiplies a vector by a numerical scalar $c$ to produce a vector ($c\mathbf{u}$). Geometrically stretches or shrinks the vector by factor $c$. Works in both $\mathbb{R}^2$ and $\mathbb{R}^3$.
  • Dot Product (Scalar Product): Multiplies two vectors to generate a single real scalar number. Works in both $\mathbb{R}^2$ and $\mathbb{R}^3$.
  • Cross Product (Vector Product): Multiplies two vectors to generate a third vector that is perpendicular (orthogonal) to both input vectors.

• Dimensional Restrictions for Cross Product:

  • Cross product operates exclusively in three-dimensional space ($\mathbb{R}^3$).
  • Two-dimensional space ($\mathbb{R}^2$) lacks the necessary dimensional space to accommodate a vector perpendicular to both input vectors.

Standard Unit Vectors and Determinant Expansion

• Standard Basis Unit Vectors in $\mathbb{R}^3$:

  • $\mathbf{i} = \langle 1, 0, 0 \rangle$
  • $\mathbf{j} = \langle 0, 1, 0 \rangle$
  • $\mathbf{k} = \langle 0, 0, 1 \rangle$

• Matrix Representation of the Cross Product:

  • For vectors $\mathbf{u} = \langle u_1, u_2, u_3 \rangle$ and $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$, the cross product $\mathbf{u} \times \mathbf{v}$ is computed using a $3 \times 3$ matrix determinant:     $\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$
  • First Row: Standard unit vectors $\mathbf{i}, \mathbf{j}, \mathbf{k}$.
  • Second Row: Components of the first vector $\mathbf{u} = \langle u_1, u_2, u_3 \rangle$.
  • Third Row: Components of the second vector $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$.

• Expansion Across First Row:

  • The determinant is expanded into three $2 \times 2$ determinants:     $\mathbf{u} \times \mathbf{v} = \mathbf{i} \begin{vmatrix} u_2 & u_3 \\ v_2 & v_3 \end{vmatrix} - \mathbf{j} \begin{vmatrix} u_1 & u_3 \\ v_1 & v_3 \end{vmatrix} + \mathbf{k} \begin{vmatrix} u_1 & u_2 \\ v_1 & v_2 \end{vmatrix}$
  • Alternating Sign Pattern:
  • First term ($\mathbf{i}$): Positive sign ($+\mathbf{i}$).
  • Second term ($\mathbf{j}$): Negative sign ($-\mathbf{j}$).
  • Third term ($\mathbf{k}$): Positive sign ($+\mathbf{k}$).

• Deleting Rows and Columns to Form $2 \times 2$ Minor Matrices:

  • Sub-matrix for $\mathbf{i}$: Eliminate Row 1 and Column 1, yielding $\begin{pmatrix} u_2 & u_3 \\ v_2 & v_3 \end{pmatrix}$.
  • Sub-matrix for $\mathbf{j}$: Eliminate Row 1 and Column 2, yielding $\begin{pmatrix} u_1 & u_3 \\ v_1 & v_3 \end{pmatrix}$.
  • Sub-matrix for $\mathbf{k}$: Eliminate Row 1 and Column 3, yielding $\begin{pmatrix} u_1 & u_2 \\ v_1 & v_2 \end{pmatrix}$.

Component Formula for Vector Cross Product

• Calculation of $2 \times 2$ Determinants:

  • For any $2 \times 2$ matrix $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$, the determinant is computed as main diagonal product minus off-diagonal product: $a \cdot d - b \cdot c$.
  • Determinant of $\mathbf{i}$ minor: $u_2 \cdot v_3 - v_2 \cdot u_3$
  • Determinant of $\mathbf{j}$ minor: $u_1 \cdot v_3 - v_1 \cdot u_3$
  • Determinant of $\mathbf{k}$ minor: $u_1 \cdot v_2 - v_1 \cdot u_2$

• Standard Vector Form:

  • Combining evaluated sub-determinants:     $\mathbf{u} \times \mathbf{v} = (u_2 \cdot v_3 - v_2 \cdot u_3)\mathbf{i} - (u_1 \cdot v_3 - v_1 \cdot u_3)\mathbf{j} + (u_1 \cdot v_2 - v_1 \cdot u_2)\mathbf{k}$

• Coordinate Vector Form:

  • Distributing the negative sign into the second component yields:     $\mathbf{u} \times \mathbf{v} = \langle u_2 \cdot v_3 - v_2 \cdot u_3, v_1 \cdot u_3 - u_1 \cdot v_3, u_1 \cdot v_2 - v_1 \cdot u_2 \rangle$

Non-Commutativity and Vector Order

• Anti-Commutative Property:

  • The cross product is anti-commutative: $\mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u})$.
  • Swapping vector order reverses the direction of the resultant vector.

• Step-by-Step Computation of $\mathbf{i} \times \mathbf{j}$:

  • Let $\mathbf{u} = \mathbf{i} = \langle 1, 0, 0 \rangle$ and $\mathbf{v} = \mathbf{j} = \langle 0, 1, 0 \rangle$.
  • First coordinate ($\mathbf{i}$ component): $u_2 \cdot v_3 - v_2 \cdot u_3 = 0 \cdot 0 - 1 \cdot 0 = 0$
  • Second coordinate ($\mathbf{j}$ component): $v_1 \cdot u_3 - u_1 \cdot v_3 = 0 \cdot 0 - 1 \cdot 0 = 0$
  • Third coordinate ($\mathbf{k}$ component): $u_1 \cdot v_2 - v_1 \cdot u_2 = 1 \cdot 1 - 0 \cdot 0 = 1$
  • Final vector: $\mathbf{i} \times \mathbf{j} = \langle 0, 0, 1 \rangle = \mathbf{k}$

• Reversal Example $\mathbf{j} \times \mathbf{i}$:

  • Let $\mathbf{u} = \mathbf{j} = \langle 0, 1, 0 \rangle$ and $\mathbf{v} = \mathbf{i} = \langle 1, 0, 0 \rangle$.
  • First coordinate: $1 \cdot 0 - 0 \cdot 0 = 0$
  • Second coordinate: $1 \cdot 0 - 0 \cdot 0 = 0$
  • Third coordinate: $0 \cdot 0 - 1 \cdot 1 = -1$
  • Final vector: $\mathbf{j} \times \mathbf{i} = \langle 0, 0, -1 \rangle = -\mathbf{k}$

Geometric Properties and Magnitude of the Cross Product

• Magnitude Formula:

  • For vectors $\mathbf{u}$ and $\mathbf{v}$ separated by angle $\theta$ ($0 \le \theta \le \pi$):     $\|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\| \cdot \|\mathbf{v}\| \cdot \sin(\theta)$
  • Contrast with Dot Product Formula:     $\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \cdot \|\mathbf{v}\| \cdot \cos(\theta)$

• Criterion for Parallel Vectors:

  • The cross product yields the zero vector $\mathbf{u} \times \mathbf{v} = \mathbf{0}$ if and only if $\sin(\theta) = 0$ (assuming non-zero vectors $\mathbf{u}$ and $\mathbf{v}$).
  • Occurs when $\theta = 0$ (identical directions) or $\theta = \pi$ ($180^\circ$, exact opposite directions).

Geometric Application: Area of Parallelograms and Triangles

• Area of a Parallelogram:

  • A parallelogram in $\mathbb{R}^3$ spanned by vectors $\mathbf{u}$ and $\mathbf{v}$ has base length $\|\mathbf{u}\|$ and height $\|\mathbf{v}\| \cdot \sin(\theta)$.
  • Area formula:     $\text{Area}_{\text{parallelogram}} = \|\mathbf{u}\| \cdot \|\mathbf{v}\| \cdot \sin(\theta) = \|\mathbf{u} \times \mathbf{v}\|$

• Area of a Triangle:

  • A triangle formed by vectors $\mathbf{u}$ and $\mathbf{v}$ occupies half the area of the corresponding parallelogram:     $\text{Area}_{\text{triangle}} = \frac{1}{2} \|\mathbf{u} \times \mathbf{v}\|$

Step-by-Step Example: Area of Triangle in 3D Space

• Problem Description:

  • Find the area of triangle $\Delta PQR$ defined by vertices:
  • $P = (0, 0, 0)$
  • $Q = (1, 2, 0)$
  • $R = (0, 1, 3)$

• Step 1: Form Vectors from Given Points

  • Vector $\mathbf{u} = \vec{PQ} = Q - P = \langle 1 - 0, 2 - 0, 0 - 0 \rangle = \langle 1, 2, 0 \rangle$
  • Vector $\mathbf{v} = \vec{PR} = R - P = \langle 0 - 0, 1 - 0, 3 - 0 \rangle = \langle 0, 1, 3 \rangle$

• Step 2: Compute Cross Product $\mathbf{u} \times \mathbf{v}$

  • Set up determinant matrix:     $\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & 0 \\ 0 & 1 & 3 \end{vmatrix}$
  • Expand matrix:
  • $\mathbf{i}$ component: $2 \cdot 3 - 1 \cdot 0 = 6$
  • $\mathbf{j}$ component: $-(1 \cdot 3 - 0 \cdot 0) = -3$
  • $\mathbf{k}$ component: $1 \cdot 1 - 0 \cdot 2 = 1$
  • Resulting vector: $\mathbf{u} \times \mathbf{v} = \langle 6, -3, 1 \rangle$

• Step 3: Compute Magnitude of Cross Product

  • $\mathbb{\|\mathbf{u} \times \mathbf{v}\|} = \sqrt{6^2 + (-3)^2 + 1^2} = \sqrt{36 + 9 + 1} = \sqrt{46}$

• Step 4: Calculate Triangle Area

  • $\text{Area}_{\text{triangle}} = \frac{1}{2} \|\mathbf{u} \times \mathbf{v}\| = \frac{\sqrt{46}}{2}$

Questions & Discussion

• Homework Volume:

  • Question: How many problems are assigned on average per section?
  • Response: Assignments average around $5$ problems per section, covering roughly $2$ sections per week.

• Vector Ordering in Cross Products:

  • Question: Does swapping the order of vectors yield the same result (e.g., using $\mathbf{j}$ as $\mathbf{u}$ and $\mathbf{i}$ as $\mathbf{v}$)?
  • Response: No, order matters. Computing $\mathbf{i} \times \mathbf{j}$ gives $\mathbf{k} = \langle 0, 0, 1 \rangle$, whereas computing $\mathbf{j} \times \mathbf{i}$ gives $-\mathbf{k} = \langle 0, 0, -1 \rangle$. Swapping vector order produces the negative vector.