Vectors & 3D GeometryTop Priority (3-4 Qs in JEE)

Vector Algebra Formulas

Dot product, cross product, scalar triple product, vector triple product, and Lagrange's identity.

7 Formulas•1,073+ Questions in Bank•

Concept Diagram

Vector Cross Product & Right-Hand Rule

Vector product a × b yielding perpendicular vector |a||b|sin(θ)n̂ and parallelogram area.

\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta, \quad \text{Projection of } \vec{a} \text{ on } \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}

Dot product yields scalar; zero if vectors are orthogonal.

Examiner Tip: Projection vector is ((a · b) / |b|²) b.

\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin\theta \, \hat{n}, \quad \text{Area of Parallelogram} = |\vec{a} \times \vec{b}|

Diagram Reference:Vector (Cross) Product

Perpendicular vector directed by right-hand thumb rule.

Examiner Tip: Area of triangle with adjacent vectors a and b is (1/2)|a × b|.

[\vec{a} \; \vec{b} \; \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}

Represents the volume of parallelepiped formed by coterminous vectors a, b, c.

Examiner Tip: Coplanarity condition: [a b c] = 0.

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

Expands double cross product into a linear combination of vectors b and c.

Examiner Tip: Memory trick: BAC minus CAB.

|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2

Connects cross product and dot product magnitudes directly.

Examiner Tip: Derived from sin²θ + cos²θ = 1.

(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = \begin{vmatrix} \vec{a}\cdot\vec{c} & \vec{a}\cdot\vec{d} \\ \vec{b}\cdot\vec{c} & \vec{b}\cdot\vec{d} \end{vmatrix}, \quad (\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}

Determinant expansion for dot product of two cross products, and linear resolution of cross product of two cross products.

Examiner Tip: Crucial identity for finding line of intersection of two planes defined by normal vectors.

\vec{a}' = \frac{\vec{b} \times \vec{c}}{[\vec{a}\,\vec{b}\,\vec{c}]}, \quad \vec{a} \cdot \vec{a}' = 1, \; \vec{a} \cdot \vec{b}' = 0, \quad V_{\text{tetrahedron}} = \frac{1}{6}|[\vec{a}\,\vec{b}\,\vec{c}]|

Reciprocal triad of vectors satisfying orthogonal dot relations, and volume of tetrahedron with coterminous edges a, b, c.

Examiner Tip: The scalar triple product of reciprocal system is [a' b' c'] = 1 / [a b c].

Frequently tested misconceptions in Vector Algebra

•Confusing vector triple product order: a × (b × c) ≠ (a × b) × c (vector cross product is not associative).
•Scalar triple product [a b c] = 0 means vectors are coplanar.