Is it correct to say: Applying cross product formula in wrong order?

a × b = −(b × a). Order matters; the sign flips.

Is it correct to say: Computing dot product using cross-product determinant?

a·b = a₁b₁ + a₂b₂ + a₃b₃ (scalar). a × b uses determinant with î, ĵ, k̂ (vector).

Is it correct to say: Confusing direction ratios (l, m, n) with direction cosines?

Direction cosines = direction ratios / magnitude of direction vector. l² + m² + n² = 1 only for cosines.

Is it correct to say: Applying skew-line distance formula to parallel lines?

For parallel lines b₁ × b₂ = 0, formula fails. Use point-to-line distance formula.

Is it correct to say: Forgetting that coplanar vectors have [a b c] = 0?

Three vectors coplanar ⟺ scalar triple product vanishes.

Is it correct to say: Using sin instead of cos for angle between two lines?

cos θ = |b₁·b₂|/(|b₁||b₂|) for lines. sin θ for line and plane using normal.

Vectors and Three-Dimensional Geometry

Vector algebra, dot product, cross product, scalar triple product, lines and planes in 3D — Maharashtra HSC Maths Ch 5 & 6

📐 12 formulas✏️ 3 examples🎯 5 practice⚖️ 8 marks🏫 Maharashtra State Board📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→Scalar triple product [a b c] = a·(b × c) as volume of parallelepiped is a 3-marker in Maharashtra Board.
→Equation of a line in vector form r = a + λb and cartesian form (x−x₀)/l = (y−y₀)/m = (z−z₀)/n — 2-marker every year.
→Angle between two planes / two lines / line and plane — three separate formulas — asked often.
→Shortest distance between two skew lines |[(a₂−a₁) b₁ b₂]|/|b₁×b₂| is a HOTS 4-marker.
→Section formula in 3D and coordinates of centroid / midpoint should be memorised.

📐 Formulas(12)

Vector Addition and Scalar Multiplication

a + b = (a_{1} + b_{1}) i^+ (a_{2} + b_{2}) j^+ (a_{3} + b_{3}) k^

Magnitude and Unit Vector★ Board fav

∣ a ∣ = a_{1}^{2} + a_{2}^{2} + a_{3}^{2}, a^= \frac{a}{∣ a ∣}

Dot Product★ Board fav

a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}

Cross Product★ Board fav

a \times b = i^a_{1} b_{1} j^a_{2} b_{2} k^a_{3} b_{3}

Area of Parallelogram / Triangle

Parallelogram: ∣ a \times b ∣; Triangle: \frac{1}{2} ∣ a \times b ∣

Scalar Triple Product★ Board fav

[a b c] = a \cdot (b \times c) = a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3}

Coplanarity Condition★ Board fav

[a b c] = 0

Vector Line Equation★ Board fav

r = a + λ b

Cartesian Line Equation

\frac{x - x _{0}}{l} = \frac{y - y _{0}}{m} = \frac{z - z _{0}}{n}

Plane in Vector Form★ Board fav

r \cdot n^= d

Angle between Two Lines / Planes

cos θ = \frac{b _{1} \cdot b _{2}}{∣ b _{1} ∣∣ b _{2} ∣} (lines)

Shortest Distance between Skew Lines★ Board fav

d = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣}

✏️ Solved Examples

1Solved Exampleeasy3 steps

Find the angle between vectors a = 2î + 3ĵ + k̂ and b = î − 2ĵ + 3k̂.

Dot product.

a \cdot b = 2 (1) + 3 (- 2) + 1 (3) = 2 - 6 + 3 = - 1

2Solved Exampleboard4 steps

Find the volume of the parallelepiped with edges a = î + ĵ + k̂, b = 2î − ĵ + 3k̂, c = î + 2ĵ − k̂.

Volume = |[a b c]|. Compute the 3×3 determinant.

[a b c] = 121 1 - 1 2 13 - 1

3Solved ExampleHOTS5 steps

Find the shortest distance between lines r = (î + 2ĵ + k̂) + λ(î − ĵ + k̂) and r = (2î − ĵ − k̂) + μ(2î + ĵ + 2k̂).

Directions: b₁ = î − ĵ + k̂, b₂ = 2î + ĵ + 2k̂. Compute b₁ × b₂.

b_{1} \times b_{2} = i^12 j^- 1 1 k^12 = - 3 i^+ 0 j^+ 3 k^

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Applying cross product formula in wrong order
✓a × b = −(b × a). Order matters; the sign flips.
2
Computing dot product using cross-product determinant
✓a·b = a₁b₁ + a₂b₂ + a₃b₃ (scalar). a × b uses determinant with î, ĵ, k̂ (vector).
3
Confusing direction ratios (l, m, n) with direction cosines
✓Direction cosines = direction ratios / magnitude of direction vector. l² + m² + n² = 1 only for cosines.
4
Applying skew-line distance formula to parallel lines
✓For parallel lines b₁ × b₂ = 0, formula fails. Use point-to-line distance formula.
5
Forgetting that coplanar vectors have [a b c] = 0
✓Three vectors coplanar ⟺ scalar triple product vanishes.
6
Using sin instead of cos for angle between two lines
✓cos θ = |b₁·b₂|/(|b₁||b₂|) for lines. sin θ for line and plane using normal.

🎯 Practice Yourself

🎯Practice Yourself5 questions

Q1
Find a unit vector perpendicular to both a = 3î + ĵ + 2k̂ and b = 2î − 2ĵ + 4k̂.
Q2
Show that vectors a = î + 2ĵ − 3k̂, b = 2î − ĵ + k̂, c = 3î + ĵ − 2k̂ are coplanar.
Q3
Find equation of the line through (1, 2, 3) in the direction (2, −1, 4) in vector form.
Q4
Find the angle between the planes 2x − y + 2z = 3 and x + 2y + 2z = 5.
Q5
Find the equation of the plane passing through (1, 2, 3) with normal 2î + 3ĵ − k̂.

📝 Notes

Vectors and 3D Geometry — Maharashtra HSC Overview

Chapters 5 and 6 of the Maharashtra Class 12 Mathematics textbook cover vector algebra (Ch 5) and lines/planes in three-dimensional geometry (Ch 6). Together they carry roughly 8 marks in the HSC paper.

Maharashtra syllabus specifics

Scalar triple product and coplanarity condition — Balbharati asks proof and numerical verification.
Vector equation of a line and plane and conversion to cartesian form — a 3-mark HSC classic.
Shortest distance between two skew lines — Maharashtra-specific 4-mark HOTS problem.
Angle between line and plane, angle between two planes — three separate formulas, all in the syllabus. Learn each separately.
Direction ratios vs direction cosines — often asked as a 2-marker with numerical.

Strategy

Start every problem by writing position vector, direction vector, or normal explicitly.
Cross product = vector; dot product = scalar. Keep the distinction clean.
Scalar triple product as determinant is FAR faster than component expansion.
For skew-line distance: check first that b₁ and b₂ are not parallel; otherwise use point-to-line distance.
Practise conversion between vector and cartesian forms — very common in HSC.