Vectors & 3D Geometry
Solve JEE Main Vectors and 3D Geometry PYQs online. Shortest distance between skew lines, scalar triple product, plane projections & 4-step solutions.
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4-Step Solution & Tricks
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π― High-Yield Syllabus Breakdown
These high-frequency concepts account for over 90% of all questions asked in Vectors & 3D Geometry:
- Dot Product, Cross Product & Vector Projections
- Scalar Triple Product [a b c] & Box Product Properties
- Vector Triple Product a Γ (b Γ c) = (aΒ·c)b - (aΒ·b)c
- Shortest Distance Between Skew Lines: |(bβ Γ bβ)Β·(aβ - aβ)| / |bβ Γ bβ|
- Lines, Direction Cosines & Coplanarity Conditions
β‘ Essential Formulas & Speed Shortcuts
Formulas and time-saving shortcuts to solve Vectors & 3D questions in under 45 seconds:
Coplanar Vectors: Three vectors are coplanar if and only if [a b c] = 0.
Projection of vector a on vector b: (a Β· b) / |b|.
Lagrangeβs Identity: |a Γ b|Β² = |a|Β² |b|Β² - (a Β· b)Β².
π Sample Solved Exam Questions
Preview authentic questions from this chapter with full step-by-step solutions:
π‘ View Step-by-Step Solution & Answer
Correct Answer: 3
- Step 1: The scalar projection of vector bβ on vector aβ is given by (bβ Β· aβ) / |aβ|.
- Step 2: Compute the dot product: bβ Β· aβ = (5)(2) + (-3)(1) + (1)(2) = 10 - 3 + 2 = 9.
- Step 3: Compute the magnitude of aβ: |aβ| = β(2Β² + 1Β² + 2Β²) = β(4 + 1 + 4) = β9 = 3.
- Step 4: Projection = 9 / 3 = 3.
β‘ Speed Tip: Projection formula: (aβ Β· bβ) / |aβ| = 9 / 3 = 3.
π‘ View Step-by-Step Solution & Answer
Correct Answer: 9/2
- Step 1: If the shortest distance between two lines is zero, the lines must intersect (they are coplanar).
- Step 2: The condition for coplanarity of (x - xβ)/aβ = (y - yβ)/bβ = (z - zβ)/cβ and (x - xβ)/aβ = (y - yβ)/bβ = (z - zβ)/cβ is:
- Step 3: |[xβ - xβ, yβ - yβ, zβ - zβ], [aβ, bβ, cβ], [aβ, bβ, cβ]| = 0.
- Step 4: |[3 - 1, k - (-1), 0 - 1], [2, 3, 4], [1, 2, 1]| = |[2, k + 1, -1], [2, 3, 4], [1, 2, 1]| = 0 β 2(3 - 8) - (k + 1)(2 - 4) - 1(4 - 3) = -10 + 2(k + 1) - 1 = 2k - 9 = 0 β k = 9/2.
β‘ Speed Tip: Coplanar condition: Setting the 3Γ3 determinant to zero gives 2k - 9 = 0 β k = 9/2.
π‘ View Step-by-Step Solution & Answer
Correct Answer: -3/2
- Step 1: Start with the vector sum: aβ + bβ + cβ = 0β.
- Step 2: Square both sides: |aβ + bβ + cβ|Β² = |0β|Β² = 0.
- Step 3: Expand the magnitude squared: |aβ|Β² + |bβ|Β² + |cβ|Β² + 2(aβ Β· bβ + bβ Β· cβ + cβ Β· aβ) = 0.
- Step 4: Since aβ, bβ, cβ are unit vectors, |aβ| = |bβ| = |cβ| = 1. Thus 1 + 1 + 1 + 2S = 0 β 3 + 2S = 0 β S = -3/2.
β‘ Speed Tip: Standard Unit Vector Identity: |aβ + bβ + cβ|Β² = 3 + 2(aβΒ·bβ + bβΒ·cβ + cβΒ·aβ) = 0 β Sum = -3/2.
β Frequently Asked Questions (FAQs)
Why is Vectors & 3D Geometry crucial for JEE Main?
Vectors & 3D carries 2 to 3 direct questions (8 to 12 marks) in every shift and has direct formula-based application.