Vectors in 3D: Dot Product, Cross Product & Triple Products

• Represent vectors in 3D using component notation
• Compute dot product and interpret it geometrically (projection, angle)
• Compute cross product and interpret it geometrically (area, perpendicular vector)
• Evaluate scalar triple product and relate it to parallelepiped volume
• Apply the BAC-CAB rule for vector triple product
• Identify coplanar vectors using the scalar triple product

Level 1 — Foundations

A vector in 3D is written as a = a₁i + a₂j + a₃k, where a₁, a₂, a₃ are scalar components along the x, y, z axes.

Magnitude: |a| = √(a₁² + a₂² + a₃²)

Unit vector: â = a/|a|

Dot product (scalar product): a·b = |a||b|cosθ, where θ is the angle between them.

• Result is a scalar.
• If a·b = 0 and neither is zero, the vectors are perpendicular.
• Projection of a on b = a·b̂ = (a·b)/|b|

Cross product (vector product): a×b is a vector perpendicular to both a and b.

• |a×b| = |a||b|sinθ
• Direction given by the right-hand rule.
• If a×b = 0 and neither is zero, the vectors are parallel.

Level 2 — JEE depth

Dot product in components: $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 = |\mathbf{a}||\mathbf{b}|\cos\theta$

$\cos\theta = \frac{a_1b_1 + a_2b_2 + a_3b_3}{|\mathbf{a}||\mathbf{b}|}$

Cross product in components (determinant form): $\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = (a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}$

$|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta \quad \text{(= area of parallelogram spanned by a and b)}$

Scalar triple product: $[\mathbf{a}\ \mathbf{b}\ \mathbf{c}] = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$

• Geometrically = volume of parallelepiped formed by a, b, c
• Coplanar condition: [a b c] = 0
• Cyclic property: [a b c] = [b c a] = [c a b]; swapping two vectors negates: [a c b] = −[a b c]

Vector triple product (BAC-CAB rule): $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$

Memory aid: BAC – CAB → keep the middle vector first.

Note: (a×b)×c = (a·c)b − (b·c)a (different from above — triple product is NOT associative).

Find the angle between a = 2i + j − k and b = i − j + 2k, and the area of the
parallelogram they span.

Step 1: Dot product
  a·b = (2)(1) + (1)(−1) + (−1)(2) = 2 − 1 − 2 = −1

Step 2: Magnitudes
  |a| = √(4+1+1) = √6
  |b| = √(1+1+4) = √6

Step 3: Angle
  cosθ = −1/(√6·√6) = −1/6
  θ = arccos(−1/6) ≈ 99.6°

Step 4: Cross product a×b
  | i   j   k  |
  | 2   1  −1  |
  | 1  −1   2  |
  = i[(1)(2)−(−1)(−1)] − j[(2)(2)−(−1)(1)] + k[(2)(−1)−(1)(1)]
  = i[2−1] − j[4+1] + k[−2−1]
  = i − 5j − 3k

Step 5: Area of parallelogram = |a×b| = √(1+25+9) = √35

• If a·b = 0, state the geometric relationship (perpendicular)
• Given two vectors, find the unit vector perpendicular to both using cross product
• Verify three vectors are coplanar using scalar triple product
• Find the area of a triangle with vertices A, B, C: Area = ½|AB × AC|
• Find volume of parallelepiped given three edge vectors

• Fig 10.14: Geometrical meaning of cross product as area of parallelogram
• Fig 10.16: Right-hand rule for direction of cross product
• The 3D axis system (x, y, z) with i, j, k unit vectors — always redraw this in exams

• a·a = ? → |a|²
• i×j = ? → k; j×k = i; k×i = j
• If [a b c] = 5, what is the volume of the parallelepiped? → 5 units³
• Is a×(b×c) = (a×b)×c? → No (verify with components)
• Stretch: Show that a×(b+c) = a×b + a×c using the determinant definition.

NCERT Chapter 10 link: Vector Algebra — covers position vectors, addition, dot and cross product through Ex 10.3 and 10.4
Exam connections: Linked to 3D geometry (lines and planes), work done by force (dot product), torque (cross product), and determinants (Chapter 4)
Study strategy: Master the 3×3 determinant expansion first; then drill 5 dot + 5 cross product problems from NCERT Misc. Exercise before attempting JEE past papers.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.