Unit 7: Vector Algebra in Space & Multi-Vector Products

Vector algebra in three-dimensional Euclidean space $\mathbb{R}^3$ forms the foundational language of mathematical physics, mechanics, and geometric analysis.

1. The Dot (Scalar Inner) Product

Given two vectors $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, their scalar product is defined geometrically as:

where $\theta \in [0, \pi]$ is the interior angle between them. In orthonormal Cartesian components:

Key Properties:

• Commutativity: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$.
• Magnitude: $|\vec{a}| = \sqrt{\vec{a} \cdot \vec{a}}$.
• Orthogonality: $\vec{a} \perp \vec{b} \iff \vec{a} \cdot \vec{b} = 0$ (for non-zero vectors).
• Cauchy-Schwarz Inequality: $|\vec{a} \cdot \vec{b}| \le |\vec{a}| |\vec{b}|$, with equality if and only if $\vec{a} \parallel \vec{b}$.

2. The Cross (Vector Outer) Product

The vector product $\vec{a} \times \vec{b}$ produces a vector perpendicular to both $\vec{a}$ and $\vec{b}$, oriented according to the right-hand rule:

where $\hat{n}$ is the unit normal vector such that $(\vec{a}, \vec{b}, \hat{n})$ forms a right-handed orthogonal triad.

In Cartesian component form:

Key Properties:

• Anti-commutativity: $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$.
• Collinearity: $\vec{a} \parallel \vec{b} \iff \vec{a} \times \vec{b} = \vec{0}$.
• Geometric Area: The magnitude $|\vec{a} \times \vec{b}|$ equals the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$.

3. Lagrange's Identity

Theorem (Lagrange's Identity): For any vectors $\vec{a}, \vec{b} \in \mathbb{R}^3$:

Proof:

From geometric definitions:

1. Definition and Determinant Representation

The scalar triple product (or box product) of three vectors $\vec{a}, \vec{b}, \vec{c} \in \mathbb{R}^3$ is defined as:

Let $\vec{a} = (a_1, a_2, a_3)$, $\vec{b} = (b_1, b_2, b_3)$, and $\vec{c} = (c_1, c_2, c_3)$. Expanding via the dot product:

This is precisely the expansion of the $3 \times 3$ determinant:

2. Geometric Interpretation: Parallelepiped and Tetrahedral Volume

• Volume of a Parallelepiped:

The base of the parallelepiped formed by $\vec{b}$ and $\vec{c}$ has area $A = |\vec{b} \times \vec{c}|$. The unit normal to the base is $\hat{n} = \frac{\vec{b} \times \vec{c}}{|\vec{b} \times \vec{c}|}$. The height $h$ is the projection of $\vec{a}$ onto $\hat{n}$: $h = |\vec{a} \cdot \hat{n}|$. Hence, the volume is:

• Volume of a Tetrahedron:

A tetrahedron with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ has volume equal to one-sixth of the parallelepiped:

3. Cyclic Symmetries and Coplanarity

From the determinant properties under row swaps:

Interchange of Dot and Cross:

• Condition of Coplanarity:

Three vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar if and only if their scalar triple product vanishes:

1. The Vector Triple Product (BAC-CAB Theorem)

Given three vectors $\vec{a}, \vec{b}, \vec{c}$, their vector triple product is $\vec{a} \times (\vec{b} \times \vec{c})$.

Theorem (The BAC-CAB Rule):

Proof:

1. Geometric Orientation: The vector $\vec{b} \times \vec{c}$ is perpendicular to the plane spanned by $\vec{b}$ and $\vec{c}$. Therefore, $\vec{a} \times (\vec{b} \times \vec{c})$ is perpendicular to $\vec{b} \times \vec{c}$, which implies it must lie in the plane of $\vec{b}$ and $\vec{c}$.

Hence, there exist scalars $\lambda, \mu$ such that:

2. Coordinate Setup: Without loss of generality, choose an orthonormal coordinate system such that:

• The $x$-axis lies along $\vec{b}$: $\vec{b} = (b_1, 0, 0)$.
• The $xy$-plane contains $\vec{c}$: $\vec{c} = (c_1, c_2, 0)$.
• $\vec{a} = (a_1, a_2, a_3)$ is general.

3. Compute the Cross Products:

Now evaluate $\vec{a} \times (\vec{b} \times \vec{c})$:

4. Compute RHS:

The LHS and RHS are identical in every component. Since the choice of coordinate axes is arbitrary, the result holds universally for all vectors in $\mathbb{R}^3$. $\blacksquare$

2. Jacobi's Identity

Summing cyclic permutations of the vector triple product:

Summing all three equations:

This is Jacobi's Identity, proving that the Lie algebra of 3D rotations $(\mathbb{R}^3, \times)$ satisfies the Jacobi relation.

3. Products of Four Vectors

1. Scalar Product of Four Vectors:

2. Vector Product of Four Vectors: