Schooling · Class 12 · Mathematics · Vector Algebra · Chapter on a page
CBSE Class 12 · Mathematics · NCERT Mathematics Part-II
Vector Algebra: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- vector
- magnitude
- equal
- Scalar multiplication
- dot product
- cross product
- Cross product in component form
- Formulas and facts
- a
- â
- Worked examples
- Finding magnitude and unit vector
- Dot product and angle between vectors
- Cross product and area of a triangle
- Watch out for
- Confusing dot and cross products
- Forgetting the minus sign in the…
- Using sin θ in the dot product…
- ×
- Treating the scalar triple product…
What this chapter is about
Vector Algebra introduces quantities that have both magnitude and direction, distinguishing them from scalars which have only magnitude. In physics and engineering, displacement, velocity, force and acceleration are vectors; temperature, mass and time are scalars. This chapter builds the algebraic machinery to add, subtract and multiply vectors in two and three dimensions.
Key ideas
- A vector is written as an ordered triple (a₁, a₂, a₃) or as a₁ i + a₂ j + a₃ k, where i, j, k are unit vectors along the x-, y- and z-axes.
- The magnitude (or modulus) of vector a = a₁ i + a₂ j + a₃ k is |a| = √(a₁² + a₂² + a₃²).
- Two vectors are equal if and only if their corresponding components are equal; a zero vector has all components zero and no definite direction.
- Scalar multiplication: k a stretches or shrinks the vector; if k < 0 the direction reverses.
- The dot product a · b = |a| |b| cos θ, where θ is the angle between the vectors; in component form a · b = a₁b₁ + a₂b₂ + a₃b₃.
- The cross product a × b is a vector perpendicular to both a and b, with magnitude |a| |b| sin θ and direction given by the right-hand rule.
- Cross product in component form: a × b = (a₂b₃ − a₃b₂) i − (a₁b₃ − a₃b₁) j + (a₁b₂ − a₂b₁) k.
- Scalar triple product [a b c] = a · (b × c) has absolute value equal to the volume of the parallelepiped formed by the three vectors; if it is zero the vectors are coplanar.
Formulas and facts to remember
- Formula / Rule: |a| = √(a₁² + a₂² + a₃²) · Meaning: Magnitude of a vector.
- Formula / Rule: Unit vector â = a / |a| · Meaning: A vector of magnitude 1 in the direction of a.
- Formula / Rule: a · b = a₁b₁ + a₂b₂ + a₃b₃ · Meaning: Dot product via components.
- Formula / Rule: a · b = |a| |b| cos θ · Meaning: Dot product via magnitudes and angle.
- Formula / Rule: a × b = (a₂b₃ − a₃b₂) i − (a₁b₃ − a₃b₁) j + (a₁b₂ − a₂b₁) k · Meaning: Cross product in components.
- Formula / Rule: |a × b| = |a| |b| sin θ · Meaning: Magnitude of cross product.
- Formula / Rule: a · a = |a|² · Meaning: Useful identity for magnitude.
- Formula / Rule: a × a = 0 · Meaning: Cross product of a vector with itself is the zero vector.
- Formula / Rule: a · b = 0 ⇔ a ⊥ b (if both non-zero) · Meaning: Perpendicularity test.
- Formula / Rule: a × b = 0 ⇔ a ∥ b (if both non-zero) · Meaning: Parallelism test.
- Formula / Rule: [a b c] = a · (b × c) · Meaning: Scalar triple product (gives volume).
Common mistakes
- Confusing dot and cross products → remember: dot gives a scalar, cross gives a vector.
- Forgetting the minus sign in the j-component when expanding a cross product determinant → write out the 3 × 3 determinant carefully every time.
- Using sin θ in the dot product formula or cos θ in the cross product magnitude → dot uses cos, cross uses sin.
- Assuming a × b = b × a → cross product is anti-commutative: a × b = −(b × a).
- Treating the scalar triple product as a vector → [a b c] is a scalar (the determinant value), not a vector.
Quick revision
- Magnitude of a = a₁ i + a₂ j + a₃ k is √(a₁² + a₂² + a₃²).
- a · b = 0 means the vectors are perpendicular (if both non-zero).
- a × b = 0 means the vectors are parallel (if both non-zero).
- Area of parallelogram with adjacent sides a, b equals |a × b|.
- Volume of parallelepiped = |a · (b × c)|.
- Cross product is anti-commutative; dot product is commutative.