Vector Algebra – JEE Main Maths Formula Sheet & Class 12 Notes | All Formulas, Products & Applications

Types of Vectors, Position Vectors, and Angle Between Vectors

Why Vector Classifications and Position Vector Relations Are the Foundation of JEE Main Vector Algebra

Scalars and Vectors (from PDF — Vector Algebra): A scalar quantity has only magnitude, no direction (mass, speed, volume, density). A vector quantity has both magnitude and direction (velocity, force, displacement, acceleration). Vectors are denoted by bold letters or with arrows; magnitude of vector a is written |a| or a.

11 Types of Vectors (from PDF — Vector Algebra JEE Main):

(1) Null (Zero) vector: Magnitude zero, direction indeterminate; denoted 0. Adding/subtracting 0 leaves any vector unchanged.

(2) Unit vector: Magnitude = 1; â = a/|a|. Used to specify direction. î, ĵ, k̂ are standard unit vectors along positive x, y, z axes.

(3) Equal vectors: a = b iff |a|=|b| AND direction same. Must satisfy both conditions.

(4) Like/Unlike vectors: Like vectors: same direction. Unlike vectors: opposite direction.

(5) Negative vector: –a has same magnitude as a but opposite direction.

(6) Collinear (parallel) vectors: Vectors whose lines of action are parallel to a fixed line. a and b are collinear iff a = λb for some scalar λ.

(7) Coplanar vectors: Vectors lying in the same plane. At least 3 coplanar non-equal vectors required to sum to zero; at least 4 non-coplanar vectors for the same.

(8) Free vectors: Not restricted to a fixed point — can be displaced parallel to themselves.

(9) Localized vectors: Restricted to a fixed point (e.g., position vectors).

(10) Co-initial vectors: Share the same initial point.

(11) Position vectors: For a fixed origin O, the position vector of P is OP = r. AB = (position vector of B) – (position vector of A) = OB – OA = b – a.

Angle between two vectors (from PDF — Vector Algebra): The angle θ between vectors a and b is the angle AOB (0 ≤ θ ≤ π) when both are drawn from the same point O. If θ = π/2 → vectors are orthogonal (perpendicular). If θ = 0 or π → vectors are parallel (or antiparallel). Download the Free PDF for Vector Algebra for all type definitions and examples for JEE Main.

Vector Addition Laws, Resultant Formulas, and Section Formula

Why Addition Laws and Section Formula Are the Most Applied Vector Algebra Basics in JEE Main

Triangle Law of Vector Addition (from PDF — Vector Algebra): If OA = a and AB = b, then OB = a + b. Three vectors are in equilibrium if they form a closed triangle in magnitude and direction (taken in order). The converse holds: vectors forming a closed polygon sum to zero (polygon law).

Parallelogram Law of Vector Addition (from PDF — Vector Algebra JEE Main): If two vectors P and Q are represented by two adjacent sides OA and OB of a parallelogram OACB, the resultant R = P + Q is represented by the diagonal OC:

Magnitude: R = √(P² + Q² + 2PQ cosθ) where θ is the angle between P and Q.

Direction: tan φ = Q sinθ / (P + Q cosθ) where φ is the angle R makes with P.

Special cases: θ=0° → R=P+Q (maximum resultant); θ=180° → R=|P–Q| (minimum); θ=90° → R=√(P²+Q²), tanφ=Q/P.

Properties of Vector Addition (from PDF — Vector Algebra):

(1) Commutative: a + b = b + a

(2) Associative: (a + b) + c = a + (b + c)

(3) Additive identity: a + 0 = 0 + a = a

(4) Additive inverse: a + (–a) = 0

Vector Subtraction (from PDF — Vector Algebra): a – b = a + (–b). Not commutative: a–b ≠ b–a (opposite direction). |a+b| ≤ |a|+|b| (triangle inequality). |a–b| ≥ ||a|–|b|| (reverse triangle inequality). If |a+b|=|a–b| → a and b are perpendicular (a·b=0).

Section Formula in Vector Algebra (from PDF — JEE Main): Let O be the origin with OA=a and OB=b. Point P dividing AB in ratio m:n:

Internal division: OP = (m·b + n·a)/(m+n) [take m times B's vector + n times A's vector, divide by m+n]

External division: OP = (m·b – n·a)/(m–n) [m≠n]

Midpoint (m=n=1): OP = (a+b)/2

Centroid of triangle with vertices a, b, c: G = (a+b+c)/3

Note from PDF: if P divides AB internally in ratio m:n, then AP:PB = m:n → OP = (m·OB + n·OA)/(m+n). Careful with the order: m multiplies the TERMINAL point B (the latter point), n multiplies the INITIAL point A. Download the Free PDF for Vector Algebra for section formula and centroid examples for JEE Main.

Linear Combination, Linearly Dependent and Independent Vectors

Why Linear Dependence Conditions Are Direct JEE Main Vector Algebra Concept Questions

Linear Combination (from PDF — Vector Algebra): Any vector r in the plane of two non-collinear vectors a and b can be uniquely written as r = xa + yb (unique x, y scalars). For three non-coplanar vectors a, b, c: any vector r = xa + yb + zc (unique x, y, z). This is the fundamental basis representation in 3D Vector Algebra.

Linearly Dependent Vectors (from PDF — Vector Algebra JEE Main): Vectors a₁, a₂, …, aₙ are linearly dependent if there exist scalars x₁, x₂, …, xₙ (not all zero) such that x₁a₁ + x₂a₂ + … + xₙaₙ = 0. Linearly independent if x₁a₁+…+xₙaₙ=0 implies x₁=x₂=…=xₙ=0.

All 7 Key Results on Linear Dependence (from PDF — Vector Algebra JEE Main):

(1) Three non-coplanar vectors a, b, c are linearly independent.

(2) If a=a₁î+a₂ĵ+a₃k̂, b=b₁î+b₂ĵ+b₃k̂, c=c₁î+c₂ĵ+c₃k̂ are linearly dependent, then the determinant det[aᵢ bᵢ cᵢ] = 0.

(3) Three non-zero vectors α₁a+β₁b+γ₁c, α₂a+β₂b+γ₂c, α₃a+β₃b+γ₃c (where a,b,c are non-coplanar) are coplanar iff det[αᵢ βᵢ γᵢ] = 0.

(4) Two non-zero, non-collinear vectors are linearly independent.

(5) Two collinear vectors, or any three coplanar vectors, or any four vectors in 3D are linearly dependent.

(6) Three points with position vectors a, b, c are collinear iff there exist scalars x, y, z (not all zero) with xa+yb+zc=0 and x+y+z=0.

(7) Four points with position vectors a, b, c, d are coplanar iff there exist scalars x, y, z, w (not all zero) with xa+yb+zc+wd=0 and x+y+z+w=0.

Scalar multiple of a vector (from PDF — Vector Algebra): m·a has magnitude |m|·|a| and same direction if m>0, opposite if m<0. Properties: (m+n)a = ma+na; m(a+b) = ma+mb; m(na) = (mn)a. Download the Free PDF for Vector Algebra for all linear dependence worked examples for JEE Main.

Scalar (Dot) Product — All 12 Properties and JEE Main Applications

Why the Dot Product Formula and Its Properties Are the Most-Used Vector Algebra Tool in JEE Main

Definition of Scalar (Dot) Product (from PDF — Vector Algebra): The scalar product of vectors a and b, denoted a·b, is defined as:

a·b = |a||b| cosθ where θ is the angle between a and b (0 ≤ θ ≤ π). The result is a scalar.

All 12 Properties of Dot Product (from PDF — Vector Algebra JEE Main):

(1) Commutative: a·b = b·a

(2) Distributive: a·(b+c) = a·b + a·c

(3) Scalar multiple: (ma)·b = m(a·b) = a·(mb)

(4) θ=0 → a·b = |a||b| (like vectors, maximum dot product)

(5) θ=π → a·b = –|a||b| (unlike vectors)

(6) Unit vectors: â·b̂ = cosθ

(7) Self dot product: a·a = |a|² → |a| = √(a·a)

(8) Perpendicularity: a·b = 0 (with a≠0, b≠0) iff a ⊥ b (θ=π/2)

(9) Standard basis vectors: î·î = ĵ·ĵ = k̂·k̂ = 1; î·ĵ = ĵ·k̂ = k̂·î = 0

(10) Component formula: if a=a₁î+a₂ĵ+a₃k̂ and b=b₁î+b₂ĵ+b₃k̂, then:

a·b = a₁b₁ + a₂b₂ + a₃b₃

And: cosθ = (a₁b₁+a₂b₂+a₃b₃) / (√(a₁²+a₂²+a₃²) · √(b₁²+b₂²+b₃²))

(11) Projection of a on b: Component of a in direction of b = a·b̂ = a·b/|b|

     Vector projection of a along b: (a·b/|b|²)·b

     Component of a perpendicular to b: a – (a·b/|b|²)·b

(12) Work done: W = F·d = |F||d|cosθ where d = AB = displacement vector

Key dot product identities in Vector Algebra (from PDF — JEE Main):

|a+b|² = |a|²+2a·b+|b|²

|a–b|² = |a|²–2a·b+|b|²

(a+b)·(a–b) = |a|²–|b|²

|a+b|²+|a–b|² = 2(|a|²+|b|²) [parallelogram law for magnitudes]

|a+b| = |a–b| iff a·b=0 (diagonals of a parallelogram are equal iff it's a rectangle). Download the Free PDF for Vector Algebra for all dot product examples for JEE Main.

Vector (Cross) Product — All 11 Properties, Determinant Form, and Areas

Why the Cross Product and Its Geometric Applications Are the Most Directly Tested Vector Algebra Formulas

Definition of Vector (Cross) Product (from PDF — Vector Algebra): The vector product of a and b is defined as:

a×b = |a||b| sinθ n̂ where θ (0≤θ≤π) is the angle between a and b, and n̂ is the unit vector perpendicular to both a and b such that a, b, n̂ form a right-handed triad (right-hand screw system). The result is a vector.

All 11 Properties of Cross Product (from PDF — Vector Algebra JEE Main):

(1) Anti-commutative: a×b ≠ b×a. In fact a×b = –(b×a)

(2) Scalar multiple: (ma)×b = m(a×b) = a×(mb)

(3) Distributive: a×(b+c) = a×b + a×c

(4) Parallel vectors: a||b → a×b = 0 (θ=0 or π → sinθ=0). In particular, a×a = 0 for any vector a.

(5) Perpendicular vectors: |a×b| = |a||b| (sinθ=1 when θ=π/2).

(6) Standard basis results: î×î = ĵ×ĵ = k̂×k̂ = 0; and from the cyclic system:

   î×ĵ = k̂; ĵ×k̂ = î; k̂×î = ĵ (cyclic: i→j→k→i is positive)

   ĵ×î = –k̂; k̂×ĵ = –î; î×k̂ = –ĵ (reverse cyclic is negative)

(7) Unit vector perpendicular to a and b: n̂ = (a×b)/|a×b|

(8) sin θ from cross product: sinθ = |a×b|/(|a||b|)

(9) Determinant formula (from PDF — Vector Algebra):

If a=a₁î+a₂ĵ+a₃k̂ and b=b₁î+b₂ĵ+b₃k̂, then:

a×b = det[î ĵ k̂; a₁ a₂ a₃; b₁ b₂ b₃] = î(a₂b₃–a₃b₂) – ĵ(a₁b₃–a₃b₁) + k̂(a₁b₂–a₂b₁)

(10) Geometric areas (from PDF — Vector Algebra):

If a and b are adjacent sides of a parallelogram: Area = |a×b|

If a and b are diagonals of a parallelogram: Area = (1/2)|a×b|

Area of triangle with sides a and b: Area = (1/2)|a×b|

Area of triangle with vertices P(p), Q(q), R(r): Area = (1/2)|PQ×PR| = (1/2)|(q–p)×(r–p)|

(11) Torque (Moment of Force): M = r×F where r = position vector from point of rotation, F = force vector. |M| = |r||F|sinθ.

Lagrange's identity (from PDF — Vector Algebra JEE Main): |a×b|² = |a|²|b|² – (a·b)² (connects dot and cross product). Equivalently: (a·b)² + |a×b|² = |a|²|b|². Download the Free PDF for Vector Algebra for all cross product and area examples for JEE Main.

Scalar Triple Product — Determinant, Volume, and Coplanarity

Why [abc] = a·(b×c) Is the Most Important Three-Vector Formula in JEE Main Vector Algebra

Definition of Scalar Triple Product (from PDF — Vector Algebra): For three vectors a, b, c: the scalar triple product [abc] = a·(b×c). This is also written as (a×b)·c. The result is a scalar.

Determinant Formula for STP (from PDF — Vector Algebra JEE Main): If a=a₁î+a₂ĵ+a₃k̂, b=b₁î+b₂ĵ+b₃k̂, c=c₁î+c₂ĵ+c₃k̂, then:

[abc] = a·(b×c) = det[a₁ a₂ a₃; b₁ b₂ b₃; c₁ c₂ c₃]

All properties of Scalar Triple Product (from PDF — Vector Algebra JEE Main):

(1) Volume of parallelopiped: If a, b, c are three adjacent edges of a parallelopiped, Volume = |[abc]|

(2) Cyclic property: [abc] = [bca] = [cab] (cyclic permutations preserve value). Anti-cyclic reversal changes sign: [bac] = –[abc].

   Also: a·(b×c) = b·(c×a) = c·(a×b) (dot and cross can be interchanged in cyclic order).

(3) Scalar multiple: [m·a, b, c] = m[abc]; [a, m·b, c] = m[abc]; [a, b, m·c] = m[abc]

(4) Coplanarity condition: Three non-zero, non-collinear vectors a, b, c are coplanar iff [abc] = 0.

   Equivalently, the three vectors are coplanar iff the determinant of their components = 0.

(5) If any two vectors in the STP are equal or parallel, [abc] = 0. In particular [aac] = 0, [aba] = 0, etc.

Important STP results (from PDF — Vector Algebra JEE Main):

[î ĵ k̂] = 1 (STP of standard basis = 1, using determinant with 3×3 identity matrix)

[abc]² = det[a·a a·b a·c; b·a b·b b·c; c·a c·b c·c] (Gram's determinant)

Volume of tetrahedron with one vertex at O and adjacent edges a, b, c: V = (1/6)|[abc]|

Volume of tetrahedron with vertices P, Q, R, S: V = (1/6)|[PQ×PR·PS]| = (1/6)|[(q–p) (r–p) (s–p)]|

For tetrahedron OABC: V = (1/6)|[a b c]| where a=OA, b=OB, c=OC.

Four points A, B, C, D (with position vectors a,b,c,d) are coplanar iff [(b–a) (c–a) (d–a)] = 0. Download the Free PDF for Vector Algebra for all STP examples for JEE Main.

Vector Triple Product and Lagrange's Identity

Why the Vector Triple Product Formula Is a Direct JEE Main Vector Algebra Identity

Vector Triple Product (from PDF — Vector Algebra): The vector triple product of three vectors a, b, c is defined as a×(b×c). Unlike the scalar triple product (which is a scalar), the vector triple product is a vector lying in the plane of b and c.

BAC-CAB Rule (from PDF — Vector Algebra JEE Main):

a×(b×c) = (a·c)b – (a·b)c

Mnemonic: "BAC minus CAB" — the first vector from the bracket (b) multiplied by the dot product of the outer vector with the last (a·c), minus the last vector from the bracket (c) multiplied by the dot product of the outer with the first (a·b).

(a×b)×c = (a·c)b – (b·c)a

Note from PDF: a×(b×c) ≠ (a×b)×c in general — the vector triple product is NOT associative.

Properties and applications of Vector Triple Product (from PDF — Vector Algebra JEE Main):

a×(b×c) lies in the plane of b and c (always expressible as linear combination of b and c).

(a×b)×c lies in the plane of a and b.

a×(b×c) + b×(c×a) + c×(a×b) = 0 (Jacobi identity).

|a×(b×c)| = |a||b×c|sinθ where θ is the angle between a and b×c.

Lagrange's Identity (from PDF — Vector Algebra):

|a×b|² + (a·b)² = |a|²|b|² = (Lagrange's identity)

Equivalently: |a×b|² = |a|²|b|² – (a·b)²

For unit vectors â and b̂: |â×b̂|² = 1 – cos²θ = sin²θ ✓ (consistent with |â×b̂| = sinθ).

Vector product rules summary (from PDF — Vector Algebra): Dot product: scalar result, commutative, distributive. Cross product: vector result, anti-commutative, distributive, NOT associative. Scalar triple product: scalar result, cyclic. Vector triple product: vector result, NOT associative. BAC-CAB: a×(b×c) = (a·c)b–(a·b)c. Download the Free PDF for Vector Algebra for all triple product examples and JEE Main applications.

Applications of Vector Algebra to Geometry — Lines, Planes, Distances

Why Vector Equations of Lines and Planes Are the Highest-Mark Application of Vector Algebra in JEE Main

Vector Equation of a Straight Line (from PDF — Vector Algebra):

Case I: Line through point with position vector a and parallel to vector b:

r = a + tb (t is scalar parameter, varies over ℝ)

Case II: Line through two points with position vectors a and b:

r = a + t(b–a) equivalently r = (1–t)a + tb

Vector Equation of a Plane (from PDF — Vector Algebra):

Case I: Plane through point a, parallel to vectors b and c:

r = a + sb + tc (s, t are scalar parameters)

Or equivalently: [r – a, b, c] = 0 → [(r–a) b c] = 0

Case II: Plane through three points a, b, c:

r = a + s(b–a) + t(c–a) = (1–s–t)a + sb + tc [s+t≤1 for interior]

Or: [(r–a), (b–a), (c–a)] = 0

Case III: Plane through point a, perpendicular to vector n:

(r – a)·n = 0 equivalently r·n = a·n [normal form; n is normal to plane]

Cartesian equivalent: ax+by+cz=d where n=aî+bĵ+ck̂ and a·n=d.

Perpendicular distance of a point from a line (from PDF — Vector Algebra JEE Main):

Line passes through a and is parallel to b. Perpendicular distance of point r from this line:

d = |(r–a)×b| / |b| [magnitude of cross product divided by |b|]

Alternatively: d = √(|r–a|² – [(r–a)·b/|b|]²) [Pythagoras: subtract the projected component]

Perpendicular distance of a point from a plane (from PDF — Vector Algebra JEE Main):

Case I: Plane through a, parallel to b and c. Distance of point r from plane:

d = |[(r–a) b c]| / |b×c| = |(r–a)·(b×c)| / |b×c|

Case II: Plane through points a, b, c. Distance of point r from plane:

d = |[(r–a) (b–a) (c–a)]| / |(b–a)×(c–a)|

Case III: Plane r·n = d. Distance of origin from plane: |d|/|n|. Distance of point p from plane: |p·n – d|/|n|.

Shortest distance between two non-intersecting (skew) lines (from PDF — Vector Algebra JEE Main):

Let two skew lines pass through a and b and are parallel to c and d respectively. If PQ is the shortest distance, n = c×d (perpendicular to both lines).

SD = |[(b–a) c d]| / |c×d| = |(b–a)·(c×d)| / |c×d|

The shortest distance is along n (perpendicular to both lines). SD = 0 iff the lines are coplanar (intersecting or parallel).

Angle bisectors of two lines in Vector Algebra (from PDF): Two lines meet at A (position vector a), parallel to b and c respectively. Equation of internal bisector: r = a + t(b̂ + ĉ) = a + t(b/|b| + c/|c|). External bisector: r = a + t(b̂ – ĉ) = a + t(b/|b| – c/|c|). Download the Free PDF for Vector Algebra for all geometry applications and JEE Main examples.