Comprehensive Vector Analysis: Algebra, Geometry, and Physical Applications

Foundations of Scalars and Vectors

• Scalar quantities are physical quantities completely specified by a single number accompanied by an appropriate physical unit:
  • Examples include mass, time, temperature, energy, and electric charge.
• Vector quantities require both a numerical magnitude and a spatial direction for complete specification:
  • Examples include displacement, velocity, acceleration, force, linear momentum, angular momentum, electric field, and magnetic field.
• In elementary vector analysis, a vector is formally defined as a quantity possessing both magnitude and direction.
• Typographical and symbolic conventions:
  • Vectors are denoted using boldface type: $\mathbf{A}$.
  • The magnitude of a vector is denoted by the corresponding plain italic letter: $A$, or alternatively by absolute value brackets: $|\mathbf{A}|$.
• Geometric representation:
  • A vector is graphically depicted as a directed line segment (arrow).
  • The length of the arrow is drawn strictly proportional to the magnitude of the vector.
  • The arrow points along the direction of the vector, with the positive sense defined by the arrowhead.

Vector Addition, Subtraction, and Equilibrium

• The Triangle Law of Addition:
  • Two vectors are added geometrically by placing the tail of the second vector at the head (arrowhead) of the first vector.
  • The resultant sum vector $\mathbf{C}$ is the directed line segment extending from the tail of the first vector to the head of the second vector:     $\mathbf{C} = \mathbf{A} + \mathbf{B}$
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• Commutativity and the Parallelogram Law:
  • Constructing a parallelogram using vectors $\mathbf{A}$ and $\mathbf{B}$ as adjacent sides demonstrates that the order of addition does not alter the resultant vector:     $\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$
  • Vector addition is strictly commutative.
  • The parallelogram construction provides an equivalent geometric formulation of the triangle law.
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• Associativity of Vector Addition:
  • For any set of three vectors, grouping the terms during addition in either sequence yields the identical resultant vector $\mathbf{D}$:     $(\mathbf{A} + \mathbf{B}) + \mathbf{C} = \mathbf{A} + (\mathbf{B} + \mathbf{C})$
  • Vector addition is strictly associative.
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• Vector Subtraction:
  • The negative of a vector is defined as a vector possessing the identical magnitude but oriented in the exact reverse direction: $-\mathbf{B} = (-1)\mathbf{B}$.
  • Vector subtraction requires no independent mathematical rule, operating as addition of the negated vector:     $\mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B})$
• Physical Realisation of Vector Equilibrium:
  • A physical mass suspended from two cords joined at a junction point demonstrates the parallelogram law in the laboratory.
  • When the junction point is in static mechanical equilibrium, the vector resultant of the cord tension forces must equal in magnitude and oppose in direction the downward gravitational force:     $\mathbf{F}_1 + \mathbf{F}_2 = -\mathbf{F}_3$
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Cartesian Components, Direction Cosines, and Basis Vectors

• Position Vectors and Coordinate Projections:
  • When the tail of a vector arrow is fixed at the Cartesian origin, its head terminates at coordinates $(x_1, y_1, z_1)$, which uniquely specify the displacement vector $\mathbf{r}$.
  • Projections onto the Cartesian axes relate directly to the vector magnitude $r$ through direction angles $\alpha, \beta, \gamma$:     $x_1 = r\cos(\alpha)$$y_1 = r\cos(\beta)$$z_1 = r\cos(\gamma)$
  • The quantities $\cos(\alpha)$, $\cos(\beta)$, and $\cos(\gamma)$ are defined as the direction cosines, where $\alpha$, $\beta$, and $\gamma$ represent the angles between $\mathbf{r}$ and the positive $x$-, $y$-, and $z$-axes, respectively.
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• Fundamental Constraint on Direction Cosines:
  • Because the three orthogonal projections must reproduce the total magnitude $r = (x_1^2 + y_1^2 + z_1^2)^{1/2}$, the direction cosines are interdependent:     $\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1$
• Arbitrary Vector Resolution:
  • Any arbitrary vector $\mathbf{A}$ is resolved into Cartesian components in the identical manner:     $A_x = A\cos(\alpha)$
  • Notation caveat: The subscript $x$ in $A_x$ specifies the component aligned with the $x$-axis; it does not indicate functional dependence on the coordinate variable $x$. The component $A_x$ may itself vary spatially as a scalar field: $A_x(x, y, z)$.
• Standard Unit Vectors and Basis Properties:
  • The unit vectors $\hat{\mathbf{i}}$, $\hat{\mathbf{j}}$, and $\hat{\mathbf{k}}$ possess unit magnitude ($1$) and point along the positive $x$-, $y$-, and $z$-axes, respectively.
  • A vector is expressed as the linear combination of its components:     $\mathbf{A} = \hat{\mathbf{i}}A_x + \hat{\mathbf{j}}A_y + \hat{\mathbf{k}}A_z$
  • Vanishing vector condition: A vector equals zero if and only if every Cartesian component vanishes independently:     $\mathbf{A} = 0 \implies A_x = A_y = A_z = 0$
  • Magnitude calculation via the three-dimensional Pythagorean theorem:     $A = (A_x^2 + A_y^2 + A_z^2)^{1/2}$
  • Basis definition: The set $\lbrace\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}\rbrace$ spans three-dimensional real space ($\mathbb{R}^3$). Because none of the three vectors can be written as a linear combination of the remaining two, they are linearly independent and constitute an orthonormal basis.
• Component-Wise Vector Arithmetic:
  • Graphical construction is replaced by direct arithmetic manipulation:     $\mathbf{A} \pm \mathbf{B} = \hat{\mathbf{i}}(A_x \pm B_x) + \hat{\mathbf{j}}(A_y \pm B_y) + \hat{\mathbf{k}}(A_z \pm B_z)$
  • Worked numerical illustration:
  • Given vectors:       $\mathbf{A} = 6\hat{\mathbf{i}} + 4\hat{\mathbf{j}} + 3\hat{\mathbf{k}}$$\mathbf{B} = 2\hat{\mathbf{i}} - 3\hat{\mathbf{j}} - 3\hat{\mathbf{k}}$
  • Vector sum:       $\mathbf{A} + \mathbf{B} = (6 + 2)\hat{\mathbf{i}} + (4 - 3)\hat{\mathbf{j}} + (3 - 3)\hat{\mathbf{k}} = 8\hat{\mathbf{i}} + \hat{\mathbf{j}}$
  • Vector difference:       $\mathbf{A} - \mathbf{B} = (6 - 2)\hat{\mathbf{i}} + (4 - (-3))\hat{\mathbf{j}} + (3 - (-3))\hat{\mathbf{k}} = 4\hat{\mathbf{i}} + 7\hat{\mathbf{j}} + 6\hat{\mathbf{k}}$
• Geometric vs. Algebraic Representation:
  • The choice between the spatial arrow $\mathbf{A}$ and the ordered component triple $(A_x, A_y, A_z)$ is strictly a distinction between a geometric picture and an algebraic picture.
  • The arrow enables spatial visualization, whereas the component triple is required for quantitative, numerical calculations.

Formal Vector Space Axioms

• Definition of a Real Vector Space:
  • In formal mathematics, an ordered triple of real numbers $(x_1, x_2, x_3)$ is designated as a vector $\mathbf{x}$.
  • The complete collection of all such ordered triples forms a three-dimensional real vector space.
• Core Axiomatic Properties (for $\mathbf{x} = (x_1, x_2, x_3)$ and $\mathbf{y} = (y_1, y_2, y_3)$):
  • Equality: $\mathbf{x} = \mathbf{y}$ if and only if $x_i = y_i$ for all indices $i = 1, 2, 3$.
  • Addition: $\mathbf{x} + \mathbf{y} = \mathbf{z}$ signifies that $x_i + y_i = z_i$ for all indices $i = 1, 2, 3$.
  • Scalar Multiplication: For any real scalar $a$, $a\mathbf{x} \rightarrow (ax_1, ax_2, ax_3)$.
  • Negative Vector: The negative inverse is $-\mathbf{x} = (-1)\mathbf{x} \rightarrow (-x_1, -x_2, -x_3)$.
  • Null Vector: There exists a null vector $\mathbf{0} \rightarrow (0, 0, 0)$.
• Derived Algebraic Properties:
  • Because vector components are ordinary real numbers, the operations inherit standard algebraic laws:
  • Commutativity of addition: $\mathbf{x} + \mathbf{y} = \mathbf{y} + \mathbf{x}$
  • Associativity of addition: $(\mathbf{x} + \mathbf{y}) + \mathbf{z} = \mathbf{x} + (\mathbf{y} + \mathbf{z})$
  • Distributivity of scalar multiplication over vector addition: $a(\mathbf{x} + \mathbf{y}) = a\mathbf{x} + a\mathbf{y}$
  • Distributivity of scalar multiplication over scalar addition: $(a + b)\mathbf{x} = a\mathbf{x} + b\mathbf{x}$
  • Associativity of scalar multiplication: $(ab)\mathbf{x} = a(b\mathbf{x})$
  • Uniqueness theorems: The null vector $\mathbf{0}$ is unique within the space, and the negative vector $-\mathbf{x}$ corresponding to any specified vector $\mathbf{x}$ is unique.

The Scalar (Dot) Product

• Definition and Algebraic Formulation:
  • The scalar product (also designated as the dot product or inner product) of two vectors yields a pure scalar quantity:     $\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z = \sum_{i} A_i B_i$
  • The scalar product is commutative:     $\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$
• Geometric Formulation:
  • Geometrically, the dot product is the product of the magnitude of one vector and the scalar projection of the second vector onto the first:     $\mathbf{A} \cdot \mathbf{B} = AB\cos(\theta)$
  • In this expression, $\theta$ denotes the angle enclosed between $\mathbf{A}$ and $\mathbf{B}$, and $B\cos(\theta)$ is the projection of $\mathbf{B}$ along $\mathbf{A}$.
  • Equivalence proof: Reorienting the coordinate frame so that $\mathbf{A}$ defines the new $x$-axis gives $A_x = A$, $A_y = A_z = 0$, and $B_x = B\cos(\theta)$, reducing $\sum A_i B_i$ directly to $AB\cos(\theta)$.
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• Orthonormality of Unit Vectors:
  • Dot products of identical standard unit vectors evaluate to unity:     $\hat{\mathbf{i}} \cdot \hat{\mathbf{i}} = \hat{\mathbf{j}} \cdot \hat{\mathbf{j}} = \hat{\mathbf{k}} \cdot \hat{\mathbf{k}} = 1$
  • Dot products of mutually perpendicular standard unit vectors vanish:     $\hat{\mathbf{i}} \cdot \hat{\mathbf{j}} = \hat{\mathbf{j}} \cdot \hat{\mathbf{k}} = \hat{\mathbf{k}} \cdot \hat{\mathbf{i}} = 0$
  • Index notation: Replacing $\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}$ with subscripted unit vectors $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ consolidates these equations into the Kronecker delta ($\delta_{mn}$):     $\mathbf{e}_m \cdot \mathbf{e}_n = \delta_{mn}$
  • Any set of vectors satisfying this relation is formally designated as orthonormal.
• Physical Applications:
  • Mechanical work performed by a constant force vector $\mathbf{F}$ acting along a displacement vector $\mathbf{s}$:     $W = \mathbf{F} \cdot \mathbf{s}$
• Orthogonality and Normal Vectors:
  • If $\mathbf{A} \cdot \mathbf{B} = 0$ while neither vector has zero magnitude ($A \neq 0$ and $B \neq 0$), then $\cos(\theta) = 0$, requiring $\theta = 90^\circ$. Thus, the vectors are orthogonal (perpendicular).
  • If $\hat{\mathbf{n}}$ is a unit vector such that $\hat{\mathbf{n}} \cdot \mathbf{r} = 0$ for every vector $\mathbf{r}$ lying within the $xy$-plane, then $\hat{\mathbf{n}}$ must be perpendicular (normal) to the entire $xy$-plane.
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• Scalar Invariance and the Vector Derivation of the Law of Cosines:
  • Defining $\mathbf{C} = \mathbf{A} + \mathbf{B}$ and taking the scalar product of $\mathbf{C}$ with itself:     $\mathbf{C} \cdot \mathbf{C} = (\mathbf{A} + \mathbf{B}) \cdot (\mathbf{A} + \mathbf{B}) = \mathbf{A} \cdot \mathbf{A} + \mathbf{B} \cdot \mathbf{B} + 2\mathbf{A} \cdot \mathbf{B}$
  • Because $\mathbf{C} \cdot \mathbf{C} = C^2$ is an invariant scalar under coordinate rotations, the dot product can be isolated:     $\mathbf{A} \cdot \mathbf{B} = \frac{1}{2}(C^2 - A^2 - B^2)$
  • The right side consists exclusively of invariant scalar lengths, proving that $\mathbf{A} \cdot \mathbf{B}$ is strictly an invariant scalar.
  • Comparing this expansion with the trigonometric law of cosines:     $C^2 = A^2 + B^2 + 2AB\cos(\theta)$
  • This comparison provides an independent verification of the geometric dot product formula, or conversely, a rigorous vector derivation of the law of cosines.
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The Vector (Cross) Product

• Geometric Definition and Handedness:
  • The vector or cross product of two vectors produces an axial vector $\mathbf{C} = \mathbf{A} \times \mathbf{B}$.
  • The magnitude of the cross product is given by:     $C = AB\sin(\theta)$
  • The direction of $\mathbf{C}$ is perpendicular to the plane formed by $\mathbf{A}$ and $\mathbf{B}$, with its sense chosen such that the triad $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ forms a right-handed coordinate system.
  • The cross product is anticommutative:     $\mathbf{A} \times \mathbf{B} = -\mathbf{B} \times \mathbf{A}$
• Unit Vector Cross Products:
  • Cross products of parallel unit vectors vanish identically:     $\hat{\mathbf{i}} \times \hat{\mathbf{i}} = \hat{\mathbf{j}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}} \times \hat{\mathbf{k}} = \mathbf{0}$
  • Cyclic permutations yield positive unit vectors:     $\hat{\mathbf{i}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}}$$\hat{\mathbf{j}} \times \hat{\mathbf{k}} = \hat{\mathbf{i}}$$\hat{\mathbf{k}} \times \hat{\mathbf{i}} = \hat{\mathbf{j}}$
  • Anticyclic reversals introduce a negative sign:     $\hat{\mathbf{j}} \times \hat{\mathbf{i}} = -\hat{\mathbf{k}}$$\hat{\mathbf{k}} \times \hat{\mathbf{j}} = -\hat{\mathbf{i}}$$\hat{\mathbf{i}} \times \hat{\mathbf{k}} = -\hat{\mathbf{j}}$
• Algebraic Component Form and Determinant Expansion:
  • The Cartesian components of the resultant vector $\mathbf{C} = \mathbf{A} \times \mathbf{B}$ are defined algebraically by:     $C_x = A_y B_z - A_z B_y$$C_y = A_z B_x - A_x B_z$$C_z = A_x B_y - A_y B_x$
  • Expanding a $3 \times 3$ determinant across its top row reproduces these component equations:     $\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$
  • Worked numerical illustration:
  • Given vectors:       $\mathbf{A} = 6\hat{\mathbf{i}} + 4\hat{\mathbf{j}} + 3\hat{\mathbf{k}}$$\mathbf{B} = 2\hat{\mathbf{i}} - 3\hat{\mathbf{j}} - 3\hat{\mathbf{k}}$
  • Determinant expansion:       $\mathbf{A} \times \mathbf{B} = \hat{\mathbf{i}}[(4)(-3) - (3)(-3)] - \hat{\mathbf{j}}[(6)(-3) - (3)(2)] + \hat{\mathbf{k}}[(6)(-3) - (4)(2)]$$\mathbf{A} \times \mathbf{B} = \hat{\mathbf{i}}[-12 - (-9)] - \hat{\mathbf{j}}[-18 - 6] + \hat{\mathbf{k}}[-18 - 8]$$\mathbf{A} \times \mathbf{B} = -3\hat{\mathbf{i}} + 24\hat{\mathbf{j}} - 26\hat{\mathbf{k}}$
• Mathematical Equivalence of Geometric and Algebraic Definitions:
  • Orthogonality proof: Dotting $\mathbf{A} \times \mathbf{B}$ into $\mathbf{A}$ or $\mathbf{B}$ via algebraic components yields zero:     $\mathbf{A} \cdot (\mathbf{A} \times \mathbf{B}) = 0$$\mathbf{B} \cdot (\mathbf{A} \times \mathbf{B}) = 0$
  • This confirms that $\mathbf{C}$ is strictly perpendicular to both $\mathbf{A}$ and $\mathbf{B}$, and normal to their plane.
  • Magnitude verification using Lagrange's identity:     $(\mathbf{A} \times \mathbf{B}) \cdot (\mathbf{A} \times \mathbf{B}) = A^2 B^2 - (\mathbf{A} \cdot \mathbf{B})^2 = A^2 B^2 - A^2 B^2 \cos^2(\theta) = A^2 B^2(1 - \cos^2(\theta)) = A^2 B^2 \sin^2(\theta)$
  • Taking the square root recovers the geometric magnitude $C = AB\sin(\theta)$.
• Geometrical Meaning: Parallelogram Area:
  • In the geometric parallelogram spanned by adjacent vectors $\mathbf{A}$ and $\mathbf{B}$, taking the magnitude $A$ as the base gives a corresponding altitude (height) of $B\sin(\theta)$.
  • The magnitude of the cross product equals the surface area of this parallelogram:     $|\mathbf{A} \times \mathbf{B}| = AB\sin(\theta) = \text{area of the parallelogram}$
  • Because the vector points normal to this planar area, physical area itself can be represented and manipulated as a directed vector quantity.
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• Physical Applications:
  • Definition of orbital angular momentum $\mathbf{L}$ given position vector (radius arm) $\mathbf{r}$ and linear momentum $\mathbf{p}$:     $\mathbf{L} = \mathbf{r} \times \mathbf{p}$
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  • Relation connecting linear velocity $\mathbf{v}$ to angular velocity $\boldsymbol{\omega}$ and radius arm $\mathbf{r}$:     $\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r}$
  • Magnetic Lorentz force $\mathbf{F}_M$ exerted on an electric charge $q$ traveling with velocity $\mathbf{v}$ through a magnetic induction field $\mathbf{B}$:     $\mathbf{F}_M = q(\mathbf{v} \times \mathbf{B})$
• Non-Existence of Vector Division:
  • Mathematical division by a vector is undefined.
  • The ratio expression $\mathbf{B}/\mathbf{A}$ cannot be uniquely specified unless the two vectors are explicitly constrained to be collinear (parallel).
• Fundamental Conceptual Distinction Between Dot and Cross Products:
  • The scalar (dot) product yields a pure number and provides a quantitative measure of how parallel two vectors are.
  • The vector (cross) product yields a directional vector and provides a quantitative measure of how perpendicular two vectors are.

Triple Products

• The Triple Scalar Product:
  • The algebraic combination $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$ evaluates to a scalar.
  • Because the alternative grouping $(\mathbf{A} \cdot \mathbf{B}) \times \mathbf{C}$ would require crossing a scalar into a vector (which is mathematically undefined), parentheses may be omitted without ambiguity: $\mathbf{A} \cdot \mathbf{B} \times \mathbf{C}$.
  • Component expansion:     $\mathbf{A} \cdot \mathbf{B} \times \mathbf{C} = A_x(B_y C_z - B_z C_y) + A_y(B_z C_x - B_x C_z) + A_z(B_x C_y - B_y C_x)$
  • Determinant representation:     $\mathbf{A} \cdot \mathbf{B} \times \mathbf{C} = \begin{vmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{vmatrix}$
  • Cyclic invariance: The value of the triple scalar product is invariant under cyclic permutation of the vectors, while anticyclic permutations invert the sign:     $\mathbf{A} \cdot \mathbf{B} \times \mathbf{C} = \mathbf{B} \cdot \mathbf{C} \times \mathbf{A} = \mathbf{C} \cdot \mathbf{A} \times \mathbf{B} = -\mathbf{A} \cdot \mathbf{C} \times \mathbf{B}$
  • Dot and cross interchangeability:     $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = (\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C}$
• Geometrical Meaning: Parallelepiped Volume:
  • The vector magnitude $|\mathbf{B} \times \mathbf{C}|$ equals the area of the parallelogram forming the base of the parallelepiped constructed on vectors $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$.
  • The vector $\mathbf{B} \times \mathbf{C}$ is oriented normal to this base plane. Dotting $\mathbf{A}$ into this normal vector multiplies the base area by the perpendicular height of the parallelepiped:     $\mathbf{A} \cdot \mathbf{B} \times \mathbf{C} = \text{volume of the parallelepiped defined by } \mathbf{A}, \mathbf{B}, \mathbf{C}$
  • Handedness and sign significance: If the calculated numerical value is negative, this sign records the intrinsic left-handedness of the vector triad $\mathbf{A}, \mathbf{B}, \mathbf{C}$ rather than indicating a calculation defect.
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• Application: The Reciprocal Lattice in Crystallography:
  • Direct crystal lattice points are defined by primitive translation vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ (not necessarily mutually orthogonal):     $\mathbf{r} = n_1\mathbf{a} + n_2\mathbf{b} + n_3\mathbf{c} \quad (n_1, n_2, n_3 \in \mathbb{Z})$
  • The reciprocal lattice basis vectors are defined by:     $\mathbf{a}' = \frac{\mathbf{b} \times \mathbf{c}}{\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})}$$\mathbf{b}' = \frac{\mathbf{c} \times \mathbf{a}}{\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})}$$\mathbf{c}' = \frac{\mathbf{a} \times \mathbf{b}}{\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})}$
  • Geometric characteristics: Vector $\mathbf{a}'$ is strictly perpendicular to the plane defined by $\mathbf{b}$ and $\mathbf{c}$, and its magnitude is proportional to $1/a$.
  • Normalization and orthogonality relations:     $\mathbf{a}' \cdot \mathbf{a} = \mathbf{b}' \cdot \mathbf{b} = \mathbf{c}' \cdot \mathbf{c} = 1$$\mathbf{a}' \cdot \mathbf{b} = \mathbf{a}' \cdot \mathbf{c} = \mathbf{b}' \cdot \mathbf{a} = \mathbf{b}' \cdot \mathbf{c} = \mathbf{c}' \cdot \mathbf{a} = \mathbf{c}' \cdot \mathbf{b} = 0$
  • Physical application: The reciprocal lattice construction is essential for describing and calculating the scattering and diffraction of waves from crystal planes.
• The Triple Vector Product and the BAC–CAB Rule:
  • The triple vector product is written as $\mathbf{A} \times (\mathbf{B} \times \mathbf{C})$.
  • Non-associativity: Parentheses are mandatory, as demonstrated by the evaluation of unit vectors:     $\hat{\mathbf{i}} \times (\hat{\mathbf{i}} \times \hat{\mathbf{j}}) = \hat{\mathbf{i}} \times \hat{\mathbf{k}} = -\hat{\mathbf{j}}$$(\hat{\mathbf{i}} \times \hat{\mathbf{i}}) \times \hat{\mathbf{j}} = \mathbf{0} \times \hat{\mathbf{j}} = \mathbf{0}$
  • Planar geometry: The vector $\mathbf{A} \times (\mathbf{B} \times \mathbf{C})$ is orthogonal to $\mathbf{A}$ and to the vector $(\mathbf{B} \times \mathbf{C})$. Because $(\mathbf{B} \times \mathbf{C})$ is normal to the plane containing $\mathbf{B}$ and $\mathbf{C}$, the resulting triple product lies in the plane formed by $\mathbf{B}$ and $\mathbf{C}$, and is expressible as a linear combination of $\mathbf{B}$ and $\mathbf{C}$.
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  • The BAC–CAB Rule identity:     $\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})$
  • The BAC–CAB rule is the single most important vector identity in physics and must be memorised due to its frequent appearance in advanced theoretical derivations.