Class 12 Mathematics Chapter 10 Vector Algebra — Formulas & Key Points

A scalar is a quantity with magnitude only (temperature, mass, time), while a vector has both magnitude and direction (velocity, force, displacement). A vector is denoted by a bold letter or with an arrow overhead. The magnitude of vector a is written |a| or simply a. The zero vector 0 has magnitude zero and arbitrary direction. Unit vector â = a/|a| has magnitude 1. Position vector of point P(x, y, z) with respect to origin O is OP = xî + yĵ + zk̂, where î, ĵ, k̂ are unit vectors along x, y, z axes. Two vectors are equal if they have the same magnitude and direction, regardless of initial point. Negative of vector a is −a, having same magnitude but opposite direction. Collinear vectors lie on parallel lines; coplanar vectors lie in the same plane.

Direction cosines of a vector are the cosines of angles α, β, γ it makes with positive x, y, z axes respectively, denoted l, m, n. For vector a = a₁î + a₂ĵ + a₃k̂, direction cosines are l = a₁/|a|, m = a₂/|a|, n = a₃/|a|. The fundamental relation l² + m² + n² = 1 always holds. Direction ratios are any three numbers a, b, c proportional to direction cosines, written as a:b:c. If direction ratios are a, b, c then direction cosines are l = ±a/√(a²+b²+c²), m = ±b/√(a²+b²+c²), n = ±c/√(a²+b²+c²). Direction ratios are not unique but direction cosines are unique up to sign. This distinction is crucial in 3D geometry problems asking for line equations or angles between lines.

Addition and subtraction of vectors follow triangle law and parallelogram law. For vectors a = a₁î + a₂ĵ + a₃k̂ and b = b₁î + b₂ĵ + b₃k̂, sum a + b = (a₁+b₁)î + (a₂+b₂)ĵ + (a₃+b₃)k̂ and difference a − b = (a₁−b₁)î + (a₂−b₂)ĵ + (a₃−b₃)k̂. Scalar multiplication: if λ is a scalar, λa = λa₁î + λa₂ĵ + λa₃k̂. Properties include commutativity (a+b=b+a), associativity ((a+b)+c=a+(b+c)), and distributive laws λ(a+b)=λa+λb and (λ+μ)a=λa+μa. Section formula: position vector of point dividing join of A(a) and B(b) in ratio m:n internally is (mb+na)/(m+n), externally (mb−na)/(m−n). These are heavily used in coordinate geometry chapters.

The dot product of vectors a and b is defined as a⋅b = |a||b|cos θ, where θ is the angle between them. It is a scalar quantity. In component form, if a = a₁î + a₂ĵ + a₃k̂ and b = b₁î + b₂ĵ + b₃k̂, then a⋅b = a₁b₁ + a₂b₂ + a₃b₃. The angle between vectors: cos θ = (a⋅b)/(|a||b|). Vectors are perpendicular if a⋅b = 0, parallel if a×b = 0. Dot product is commutative (a⋅b = b⋅a) and distributive over addition (a⋅(b+c) = a⋅b + a⋅c). Special cases: î⋅î = ĵ⋅ĵ = k̂⋅k̂ = 1, î⋅ĵ = ĵ⋅k̂ = k̂⋅î = 0. Projection of a on b is (a⋅b)/|b|. Work done by force F over displacement d is W = F⋅d. The dot product appears in 4–5 mark questions testing angle calculation and perpendicularity conditions.

The cross product a×b is a vector perpendicular to both a and b, with magnitude |a×b| = |a||b|sin θ, where θ is angle between them. Direction is given by right-hand thumb rule. In component form, a×b = (a₂b₃−a₃b₂)î − (a₁b₃−a₃b₁)ĵ + (a₁b₂−a₂b₁)k̂, or determinant form |î ĵ k̂; a₁ a₂ a₃; b₁ b₂ b₃|. Cross product is anti-commutative: a×b = −(b×a). Distributive over addition: a×(b+c) = a×b + a×c. Vectors parallel if a×b = 0. Special cases: î×ĵ = k̂, ĵ×k̂ = î, k̂×î = ĵ; cyclic order forward gives positive, reverse gives negative. Area of parallelogram with adjacent sides a, b is |a×b|; triangle area is ½|a×b|. Cross product questions are standard 4-mark board problems.

Scalar triple product [a b c] = a⋅(b×c) gives the volume of parallelepiped with edges a, b, c. In component form, it equals the determinant |a₁ a₂ a₃; b₁ b₂ b₃; c₁ c₂ c₃|. Cyclic permutation does not change value: [a b c] = [b c a] = [c a b], but anti-cyclic or swapping two vectors changes sign: [a c b] = −[a b c]. Vectors a, b, c are coplanar if and only if [a b c] = 0. Volume of tetrahedron with edges a, b, c is (1/6)|[a b c]|. Vector triple product identities: a×(b×c) = (a⋅c)b − (a⋅b)c (BAC−CAB rule) and (a×b)×c = (a⋅c)b − (b⋅c)a. These identities simplify complex vector proofs and appear in 5–6 mark derivation questions in CBSE exams and competitive tests like JEE Advanced.

Vector Algebra provides elegant formulas for geometric quantities. Area of triangle ABC with position vectors a, b, c is ½|AB×AC| = ½|(b−a)×(c−a)|. Area of parallelogram is |a×b| where a and b are adjacent sides. Projection of vector a on b is scalar (a⋅b)/|b|; vector form is ((a⋅b)/|b|²)b. Work done by constant force F displacing object by d is W = F⋅d. Moment of force F about point O with position vector r is τ = r×F. Distance from point P(r) to line through A(a) parallel to b is |AP×b|/|b| = |(r−a)×b|/|b|. These applications are favorite topics for CBSE board long-answer questions worth 6 marks, often combined with coordinate geometry, especially in March board exams where 3D geometry integrates with vectors extensively.

Understanding properties avoids calculation errors. Dot product is commutative (a⋅b=b⋅a), distributive (a⋅(b+c)=a⋅b+a⋅c), and satisfies a⋅a=|a|². It does not have an inverse or cancellation law. Cross product is anti-commutative (a×b=−b×a), distributive over addition, satisfies a×a=0, but is not associative: (a×b)×c ≠ a×(b×c). Jacobi identity: a×(b×c) + b×(c×a) + c×(a×b) = 0. For orthonormal basis î, ĵ, k̂: î⋅î=1, î×ĵ=k̂, etc. Lagrange identity: |a×b|² = |a|²|b|² − (a⋅b)². These properties simplify proofs in vector identities and 5-mark 'prove that' questions where you must show a relation between scalar and vector products, common in CBSE sample papers and previous years' board exams.

Students often lose marks due to preventable errors in Vector Algebra. Confusing dot and cross products is the most common mistake: dot product yields a scalar, cross product a vector. Sign errors plague cross product calculation—always expand the determinant carefully, and remember the middle term î coefficient has a negative sign. Forgetting the unit vector n̂ in a×b = |a||b|sin θ n̂ leads to dimensional confusion. Writing a⋅b = |a||b| without cos θ is incorrect. Mixing up cyclic order in î×ĵ=k̂, ĵ×k̂=î, k̂×î=ĵ versus reverse order (anti-cyclic gives negative sign) causes sign flips. In scalar triple product, swapping two vectors changes sign, not detected if you compute determinant carelessly. Direction cosines must satisfy l²+m²+n²=1; if your answer does not, recheck magnitude calculation. Finally, stating vectors are parallel when a⋅b=0 (correct condition: a×b=0) and perpendicular when a×b=0 (correct: a⋅b=0) inverts the logic—mark these clearly in your formula sheet to avoid last-minute panic.

Mnemonic for cyclic cross product order: 'I Jump Constantly' → î×ĵ=k̂, ĵ×k̂=î, k̂×î=ĵ. Reverse gives negative. Remember BAC−CAB rule for vector triple product a×(b×c) as 'Back of the Cab': (a⋅c)b − (a⋅b)c. Dot product uses 'Cosine', Cross product uses 'Sine'—C and D (Dot), S and C (Cross) alphabetically. For determinant expansion: 'î positive, ĵ negative, k̂ positive' matches the (-1)^(i+j) pattern. Projection scalar is (a⋅b)/|b|; vector projection adds b̂ unit vector → ((a⋅b)/|b|²)b. Remember direction cosines constraint as 'unit sphere': l²+m²+n²=1. Coplanarity check: 'zero determinant = zero volume = coplanar'. CBSETUTOR.ai offers 24×7 AI tutoring where students upload formula-sheet photos and get instant clarifications on these mnemonics, worked step-by-step, ensuring you never forget under exam pressure—just ₹999/month for Classes 6-12 with a 3-day free trial to test before boards.

Example 1: Find unit vector perpendicular to both a=î−ĵ+2k̂ and b=2î+3ĵ−k̂. Compute a×b = |î ĵ k̂; 1 −1 2; 2 3 −1| = î(1−6)−ĵ(−1−4)+k̂(3+2) = −5î+5ĵ+5k̂. Magnitude |a×b|=√(25+25+25)=5√3. Unit vector = (−5î+5ĵ+5k̂)/(5√3) = (−î+ĵ+k̂)/√3. Example 2: Show vectors a=2î+ĵ−2k̂, b=î+ĵ, c=î−2ĵ+3k̂ are coplanar. [a b c]=det|2 1 −2; 1 1 0; 1 −2 3|=2(3−0)−1(3−0)−2(−2−1)=6−3+6=9≠0, so not coplanar (correction: they are not). Example 3: Find projection of a=3î+4k̂ on b=4î+3k̂. a⋅b=12+12=24, |b|=√(16+9)=5. Projection = 24/5. These examples mirror typical CBSE 4-mark board questions and help cement formula application under timed conditions, crucial for scoring 30+ in this chapter during March 2025 exams.

This quick-reference box consolidates every formula you must recall in the exam hall. Magnitude: |a|=√(a₁²+a₂²+a₃²). Unit vector: â=a/|a|. Direction cosines l,m,n: l²+m²+n²=1. Dot product: a⋅b=a₁b₁+a₂b₂+a₃b₃=|a||b|cos θ; perpendicular if a⋅b=0. Cross product: a×b=(a₂b₃−a₃b₂)î−(a₁b₃−a₃b₁)ĵ+(a₁b₂−a₂b₁)k̂=|a||b|sin θ n̂; parallel if a×b=0. Scalar triple product: [a b c]=a⋅(b×c); coplanar if =0. Volume parallelepiped=|[a b c]|, tetrahedron=(1/6)|[a b c]|. Area triangle=½|AB×AC|, parallelogram=|a×b|. Projection of a on b: (a⋅b)/|b|. Vector triple product: a×(b×c)=(a⋅c)b−(a⋅b)c. Keep this box on your desk during final revision week; practice writing it from memory daily. CBSETUOR.ai students upload this box and drill with AI-generated questions, tracking weak spots till exam day—affordable at ₹999/month, one subscription covering all subjects Classes 6-12 with photo-upload doubt solving anytime, anywhere.