Vector Algebra & Multi-Products

Vector Space Axioms

Vectors in 3D Euclidean space $R^{3}$ possess magnitude and direction, obeying parallelogram addition and scalar scaling. In IIT-JEE Advanced, mastery requires fluency in vector triple products and vector differential/algebraic equations.

1. Dot & Cross Products

📊Visual Mathematical Intuition

Vector Multi-Products: Cross Product Normal & Parallelepiped Volume

Cross Product Normal:

\vec{c} = \vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \, \hat{n} \implies \vec{c} \perp \vec{a} \text{ & } \vec{c} \perp \vec{b}

Scalar Triple Product:

[\vec{a} \; \vec{b} \; \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = \text{Volume of Parallelepiped}

Vector Triple Product:

\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} \text{ (BAC-CAB)}

Vector cross product $\vec{a} \times \vec{b}$ spanning normal $\vec{c} \perp \vec{a}, \vec{b}$ , and scalar triple product volume of a parallelepiped $[\vec{a} \; \vec{b} \; \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})$ .

Scalar (Dot) Product

\vec{a} \cdot \vec{b} = | \vec{a} | | \vec{b} | \cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}

Projection of $\vec{a}$ onto $\vec{b}$ : ${Proj}_{\vec{b}} (\vec{a}) = \frac{\vec{a} \cdot \vec{b}}{| \vec{b} |}$
Orthogonality: $\vec{a} ⊥ \vec{b} ⟺ \vec{a} \cdot \vec{b} = 0$

Vector (Cross) Product

\vec{a} \times \vec{b} = | \vec{a} | | \vec{b} | \sin θ \hat{n} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix} |

Area of Parallelogram formed by $\vec{a}, \vec{b}$ : $Area = | \vec{a} \times \vec{b} |$
Collinearity: $\vec{a} ∥ \vec{b} ⟺ \vec{a} \times \vec{b} = \vec{0}$

2. Scalar Triple Product (Box Product $[\vec{a} \vec{b} \vec{c}]$ )

[\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = | \begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} |

Properties of Box Product

Geometric Meaning: Volume of parallelopiped with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is $V = | [\vec{a} \vec{b} \vec{c}] |$ . Volume of tetrahedron is $\frac{1}{6} | [\vec{a} \vec{b} \vec{c}] |$ .
Cyclic Permutation: $[\vec{a} \vec{b} \vec{c}] = [\vec{b} \vec{c} \vec{a}] = [\vec{c} \vec{a} \vec{b}] = - [\vec{b} \vec{a} \vec{c}]$ .
Coplanarity Criterion: $\vec{a}, \vec{b}, \vec{c}$ are coplanar iff $[\vec{a} \vec{b} \vec{c}] = 0$ .

3. Vector Triple Product (BAC - CAB Rule)

BAC - CAB Expansion Rule

\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}

(\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{b} \cdot \vec{c}) \vec{a}

Vector triple product is NOT associative: $\vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c}$ .

4. Scalar & Vector Products of Four Vectors

Lagrange's Identity: $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = (\vec{a} \cdot \vec{c}) (\vec{b} \cdot \vec{d}) - (\vec{a} \cdot \vec{d}) (\vec{b} \cdot \vec{c}) = | \begin{matrix} \vec{a} \cdot \vec{c} & \vec{a} \cdot \vec{d} \\ \vec{b} \cdot \vec{c} & \vec{b} \cdot \vec{d} \end{matrix} |$
Vector Product of Four Vectors: $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = [\vec{a} \vec{b} \vec{d}] \vec{c} - [\vec{a} \vec{b} \vec{c}] \vec{d} = [\vec{a} \vec{c} \vec{d}] \vec{b} - [\vec{b} \vec{c} \vec{d}] \vec{a}$

Vector Algebra & Multi-Products ​

1. Dot & Cross Products ​

Scalar (Dot) Product ​

Vector (Cross) Product ​

2. Scalar Triple Product (Box Product [a→b→c→]) ​

Properties of Box Product ​

3. Vector Triple Product (BAC - CAB Rule) ​

4. Scalar & Vector Products of Four Vectors ​