🎯 Key Points
- Dot product gives a SCALAR (a·b=0 → perpendicular); Cross product gives a VECTOR (a×b=0 → parallel) — opposite tests for opposite relationships, easy to mix up
- |a×b| = area of the PARALLELOGRAM spanned by a,b; half that = area of the triangle — a fast geometric shortcut avoiding the base×height formula
- Scalar triple product [a b c] = a·(b×c) = volume of the parallelepiped; equals 0 exactly when a,b,c are COPLANAR
- Vector triple product (BAC-CAB rule): a×(b×c) = (a·c)b−(a·b)c — memorize this exact pattern, re-deriving it from scratch each time wastes exam time
Vector Algebra
A vector has both magnitude and direction. Used to represent forces, velocities, and geometric relationships in space.
Types of Vectors
- Zero vector: Magnitude 0, no definite direction
- Unit vector: Magnitude 1; a_hat = a / |a|
- Position vector: Vector from origin O to point P
- Collinear: Parallel vectors; one is a scalar multiple of the other
- Coplanar: Scalar triple product = 0
Operations
Triangle law of vector addition: placing vector b at the head of vector a, the diagonal from the start to the final head gives the resultant a + b.
- Addition (Triangle law): Place head to tail; closing vector = sum
- Section formula (internal): r = (mb + na)/(m + n) for ratio m:n
- Midpoint: (a + b)/2
Dot Product (Scalar Product)
- a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃
- a · b = 0 ⟹ perpendicular; a · b = |a||b| ⟹ parallel
- Scalar projection of a onto b: (a · b) / |b|
- i·i = j·j = k·k = 1; i·j = j·k = k·i = 0
Cross Product (Vector Product)
- |a × b| = |a||b| sin θ (direction by right-hand rule)
- Anti-commutative: a × b = −b × a
- a × b = 0 ⟹ vectors are parallel
- Area of parallelogram = |a × b|; triangle = ½|a × b|
- i × j = k, j × k = i, k × i = j (cyclic order)
Scalar Triple Product [a b c]
a · (b × c) = volume of parallelepiped. [a b c] = 0 ⟹ coplanar.
Cyclic: [a b c] = [b c a] = [c a b]; swapping two negates.
Vector Triple Product
a × (b × c) = (a·c)b − (a·b)c (BAC-CAB rule)
🚀 JEE Advanced Edge
Why the scalar triple product test for coplanarity is preferred over checking pairwise parallelism: Three vectors a,b,c are coplanar exactly when [a b c]=0, a single determinant computation — this is far more reliable than trying to verify coplanarity by inspection or pairwise relationships, especially when none of the three vectors are individually parallel to each other but together they still happen to lie in one plane.
Lagrange's identity connects dot and cross products: |a×b|² = |a|²|b|² − (a·b)² — this identity lets you find the cross product's magnitude purely from dot products and magnitudes, without computing the cross product component-wise, useful when a problem only gives you magnitudes and the angle (or dot product) between vectors.
Worked problem: If |a|=3, |b|=4, and a·b=6, find |a×b|. Approach: Using Lagrange's identity: |a×b|² = |a|²|b|² − (a·b)² = 9×16 − 36 = 144−36 = 108. So |a×b| = √108 = 6√3.
Worked Example: Dot Product and Angle Between Vectors
Find the angle between a = 2î + 2ĵ − k̂ and b = î − 2ĵ + 2k̂.
a · b = (2)(1) + (2)(−2) + (−1)(2) = 2 − 4 − 2 = −4. |a| = √(4+4+1) = 3. |b| = √(1+4+4) = 3.
cos θ = (a·b)/(|a||b|) = −4/9. θ = cos⁻¹(−4/9) ≈ 116.4°. The vectors are obtuse to each other (dot product negative → angle > 90°). This quick check avoids computing the full inverse cosine in MCQs.
Worked Example: Cross Product and Area of a Parallelogram
Find the area of the parallelogram with adjacent sides a = î + 2ĵ and b = 3î − k̂.
a × b = |î ĵ k̂ / 1 2 0 / 3 0 −1| = î(2·(−1)−0·0) − ĵ(1·(−1)−0·3) + k̂(1·0−2·3) = −2î + ĵ − 6k̂.
Area = |a × b| = √(4 + 1 + 36) = √41. Area of parallelogram = |a × b|; area of triangle with same sides = ½|a × b|.
Components of a Vector and Magnitude
- In 3D any vector is written in component form r = xî + yĵ + zk̂, where î, ĵ, k̂ are unit vectors along the axes.
- The position vector of P(x, y, z) is OP = xî + yĵ + zk̂ with magnitude |r| = √(x² + y² + z²).
- Vector joining A(x₁,y₁,z₁) to B(x₂,y₂,z₂): AB = (x₂−x₁)î + (y₂−y₁)ĵ + (z₂−z₁)k̂.
- Addition and scalar multiplication act component-wise; the unit vector along r is r̂ = r / |r|.
Direction Cosines and Direction Ratios of a Vector
- The direction cosines of r = aî + bĵ + ck̂ are l = a/|r|, m = b/|r|, n = c/|r|, the cosines of the angles r makes with the x, y, z axes.
- They satisfy the identity l² + m² + n² = 1.
- The components a, b, c themselves are the direction ratios — any set proportional to (l, m, n).
- From direction ratios, direction cosines are a/√(a²+b²+c²), b/√(a²+b²+c²), c/√(a²+b²+c²).
Section Formula (Internal and External Division)
- Internal division: the point dividing the segment joining a and b in ratio m:n has position vector (m b + n a)/(m + n).
- External division: the corresponding point is (m b − n a)/(m − n).
- Midpoint (ratio 1:1): (a + b)/2.
- These mirror the coordinate section formulae, applied to position vectors directly.
Projection of a Vector
- Scalar projection of a on b: (a·b)/|b| = |a| cos θ (a signed length; negative when the angle is obtuse).
- Vector projection of a on b: ((a·b)/|b|²) b — the component of a along the direction of b.
- The projection of a on a unit vector b̂ is simply a·b̂.
- Any vector splits into a component parallel to b (its projection) and a component perpendicular to b.
Applications: Work Done and Torque
- Work done by a constant force F over displacement d is the scalar (dot) product W = F·d = |F||d| cos θ.
- Moment (torque) of a force F acting at position r about a point is the vector (cross) product τ = r × F; its magnitude is |r||F| sin θ.
- Dot product suits scalar quantities (work, projection); cross product suits quantities with a rotational sense (torque, angular momentum, area).
- The area of a triangle with adjacent sides a and b is ½|a × b|.