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MathsMediumClass 12

Vectors — Dot Cross Triple Product

Vectors

52

JEE Qs

18%

Hard

90

min

Focus on both geometric interpretations and algebraic manipulation of vector products to tackle a wide range of problem types efficiently.

🧮 Key Formulas

a . b = |a||b|cosθ
a . b = a_x b_x + a_y b_y + a_z b_z
Projection of vector a onto vector b = (a . b) / |b|
|a x b| = |a||b|sinθ
a x b = i(a_y b_z - a_z b_y) - j(a_x b_z - a_z b_x) + k(a_x b_y - a_y b_x)
Area of parallelogram with adjacent sides a, b = |a x b|
Area of triangle with adjacent sides a, b = (1/2)|a x b|
Scalar Triple Product (STP): [a b c] = a . (b x c)
STP in components: [a b c] = | a_x a_y a_z | | b_x b_y b_z | | c_x c_y c_z | (determinant)
Volume of parallelepiped with co-terminal edges a, b, c = |[a b c]|
Volume of tetrahedron with co-terminal edges a, b, c = (1/6)|[a b c]|
Condition for coplanarity of vectors a, b, c: [a b c] = 0
Vector Triple Product (VTP): a x (b x c) = (a . c)b - (a . b)c
Lagrange's Identity: (a x b) . (c x d) = (a . c)(b . d) - (a . d)(b . c)
(a x b)^2 = |a|^2|b|^2 - (a . b)^2

✅ Key Points for JEE

  • 1Master the geometric interpretations: Dot product for angle and projection, Cross product for area and perpendicular vector, STP for volume and coplanarity.
  • 2Pay attention to the scalar vs. vector nature of the results: Dot product and STP yield scalars; Cross product and VTP yield vectors.
  • 3Understand the algebraic properties: Dot product is commutative and distributive; Cross product is anti-commutative and distributive; STP is cyclic in vector order, and VTP uses the BAC-CAB rule for expansion.
  • 4For complex vector identities, strategically use the VTP expansion (BAC-CAB rule) and Lagrange's identity. Convert scalar products like (a x b)^2 into (a x b).(a x b) for simplification.
  • 5Always check for coplanarity conditions by setting the Scalar Triple Product to zero. This is a very common application in JEE.

⚠️ Common Mistakes

  • ✕Confusing the order of vectors in cross products (a x b ≠ b x a) and VTP (a x (b x c) ≠ (a x b) x c).
  • ✕Errors in calculating determinants for cross product or Scalar Triple Product, especially with signs.
  • ✕Incorrectly applying the BAC-CAB rule for VTP, leading to algebraic errors in expanded expressions.
  • ✕Forgetting to take the absolute value for volumes and areas derived from vector products, as these quantities must be non-negative.

📝 Practice Questions

See all

Q10.Let →a and →b be two unit vectors such that the angle between them is . If λ→a + 2→b and 3→a −λ→b are 3 perpendicular to each other, then the number of values of λ in [−1, 3] is : (1) 2 (2) 1 (3) 0 (4) 3 e 1 x x loge α

2025·MCQEasy

Q10.Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC , divides the arc −−−length of arc AB 1 → → → AC such that length of arc BC = 5 , and OC = αOA + βOB , then α + √2(√3 −1)β is equal to (1) 2√3 (2) 2 −√3 (3) 5√3 (4) 2 + √3 f ∘g is

2025·Multi conceptHard

Q23.Let →c be the projection vector of →b = λ^i + 4^k, λ > 0, on the vector →a = ^i + 2^j + 2^k. If |→a + →c| = 7, then the area of the parallelogram formed by the vectors →b and →c is ________

2025·NumericalMedium

Q4. Let a line pass through two distinct points P(−2, −1, 3) and Q , and be parallel to the vector 3^i + 2^j + 2^k. If the distance of the point Q from the point R(1, 3, 3) is 5 , then the square of the area of △PQR is equal to : (1) 148 (2) 136 (3) 144 (4) 140

2025·MCQMedium

Q12.Let →a = 3^i −^j + 2^k, b =→a× (^i −2^k) and→c= b × ^k. Then the projection of→c−2^j on →a is : (1) 2√14 (2) √14 (3) 3√7 (4) 2√7

2025·MCQMedium

Q48.Two particles are located at equal distance from origin. The position vectors of those are represented by ¯A = 2^i + 3n^j + 2^k and ¯B = 2^i −2^j + 4p^k, respectively. If both the vectors are at right angle to each other, the value of n−1 is _____ .

2025·NumericalEasy

NCERT Chapters

  • Class 12 Maths Ch 10: Vectors