Board exam focus — Vector Algebra (CBSE & HBSE)

Vector Algebra carries 8 marks in CBSE. Topics include types of vectors, vector operations, dot and cross products. Questions typically involve finding unit vectors, dot/cross products, angle between vectors, and projection.

Types of Vectors and Vector Operations

Types of Vectors

Representation

Vector a⃗ = a₁î + a₂ĵ + a₃k̂ Magnitude: |a⃗| = √(a₁²+a₂²+a₃²) Unit vector: â = a⃗/|a⃗| = (a₁î+a₂ĵ+a₃k̂)/|a⃗|

Section Formula

Point P dividing AB in ratio m:n (internally): OP⃗ = (m·OB⃗ + n·OA⃗)/(m+n)

Midpoint of AB: (OA⃗ + OB⃗)/2

Vector Addition Properties

• a⃗ + b⃗ = b⃗ + a⃗ (commutative)
• (a⃗+b⃗)+c⃗ = a⃗+(b⃗+c⃗) (associative)
• a⃗ + 0⃗ = a⃗ (additive identity)
• a⃗ + (−a⃗) = 0⃗ (additive inverse)

Triangle Law: If AB⃗ = a⃗ and BC⃗ = b⃗, then AC⃗ = a⃗ + b⃗. Parallelogram Law: Diagonal = sum of two adjacent sides.

CBSE Tip: Always express vectors in component form (a₁î+a₂ĵ+a₃k̂) for computations. For unit vectors, divide by magnitude. The section formula is often tested with 3D coordinates.

Dot Product (Scalar Product)

Dot Product

Definition: a⃗·b⃗ = |a⃗||b⃗|cosθ (where θ is the angle between them)

Component form: If a⃗ = a₁î+a₂ĵ+a₃k̂ and b⃗ = b₁î+b₂ĵ+b₃k̂: a⃗·b⃗ = a₁b₁ + a₂b₂ + a₃b₃

Key Results:

• î·î = ĵ·ĵ = k̂·k̂ = 1
• î·ĵ = ĵ·k̂ = k̂·î = 0
• a⃗·b⃗ = 0 ⟺ a⃗ ⊥ b⃗ (for non-zero vectors)

Properties:

• a⃗·b⃗ = b⃗·a⃗ (commutative)
• a⃗·(b⃗+c⃗) = a⃗·b⃗ + a⃗·c⃗ (distributive)
• (ka⃗)·b⃗ = k(a⃗·b⃗)
• a⃗·a⃗ = |a⃗|²

Angle Between Two Vectors

cosθ = a⃗·b⃗/(|a⃗||b⃗|)

Projection

Projection of a⃗ on b⃗ = (a⃗·b⃗)/|b⃗| Vector projection of a⃗ on b⃗ = [(a⃗·b⃗)/|b⃗|²]·b⃗

Work Done

W = F⃗·d⃗ = |F⃗||d⃗|cosθ

CBSE Tip: To check if vectors are perpendicular, compute the dot product and verify it equals zero. To find the angle between vectors, use cosθ = a⃗·b⃗/(|a⃗||b⃗|). The projection formula is tested in 2-mark questions.

Cross Product and Scalar Triple Product

Cross Product (Vector Product)

Definition: a⃗×b⃗ = |a⃗||b⃗|sinθ n̂ (where n̂ is unit vector perpendicular to both, direction by right-hand rule)

Component form (determinant formula): a⃗×b⃗ = |î ĵ k̂; a₁ a₂ a₃; b₁ b₂ b₃| = (a₂b₃−a₃b₂)î − (a₁b₃−a₃b₁)ĵ + (a₁b₂−a₂b₁)k̂

Key Results:

• î×î = ĵ×ĵ = k̂×k̂ = 0⃗
• î×ĵ = k̂, ĵ×k̂ = î, k̂×î = ĵ
• ĵ×î = −k̂, k̂×ĵ = −î, î×k̂ = −ĵ
• a⃗×b⃗ = 0⃗ ⟺ a⃗ ∥ b⃗ (or one is zero)

Properties:

• a⃗×b⃗ = −(b⃗×a⃗) (NOT commutative; anti-commutative)
• a⃗×(b⃗+c⃗) = a⃗×b⃗ + a⃗×c⃗ (distributive)

Geometric Applications

• Area of parallelogram (with adjacent sides a⃗ and b⃗): |a⃗×b⃗|
• Area of triangle (with adjacent sides a⃗ and b⃗): (1/2)|a⃗×b⃗|

Scalar Triple Product

[a⃗ b⃗ c⃗] = a⃗·(b⃗×c⃗) = det([[a₁,a₂,a₃],[b₁,b₂,b₃],[c₁,c₂,c₃]])

Volume of parallelepiped with edges a⃗, b⃗, c⃗ = |[a⃗ b⃗ c⃗]|

Coplanarity: a⃗, b⃗, c⃗ are coplanar ⟺ [a⃗ b⃗ c⃗] = 0

CBSE Tip: The area of a parallelogram/triangle using cross product is a standard 3-mark question. Use the determinant formula for the cross product to avoid errors.

Frequently asked questions

Are these Vector Algebra notes free?

Yes — the Vector Algebra notes for Mathematics (Class 12) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Vector Algebra notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Vector Algebra chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Vector Algebra.