The topic-specific trap you need to name: confusing the scalar (dot) product with the vector (cross) product — mixing up which one yields a scalar and which yields a vector, and misapplying the angle-dependent factor (cos θ vs sin θ).
The scalar product (dot product) of two vectors A and B is defined as A · B = AB cos θ, where θ is the angle between them. The result is a scalar — a pure number with no direction. It measures how much one vector projects along the other. When the vectors are perpendicular (θ = 90°), the dot product is zero. When parallel, it equals the product of magnitudes.
The vector product (cross product) is defined as A × B = AB sin θ n̂, where n̂ is a unit vector perpendicular to the plane of A and B, determined by the right-hand rule. The result is a vector. Its magnitude equals the area of the parallelogram formed by A and B. When parallel (θ = 0° or 180°), the cross product is zero. When perpendicular, magnitude is maximum.
Key distinctions (NCERT Class 11 Physics Chapter 3, page 27):
Watch-out for NEET: questions test whether you apply cos θ (dot) or sin θ (cross), and whether you correctly identify the result type. The angle-reference trap — whether θ is measured from one vector or from the perpendicular — catches aspirants under time pressure. Always confirm: dot → cos → scalar; cross → sin → vector → right-hand rule for direction.
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
The scalar product A · B of two vectors is equal to:
Answer: C. By definition, the scalar (dot) product of two vectors A and B is AB cos θ, where θ is the angle between them (NCERT Class 11 Physics Chapter 3, page 27).
Why A is wrong: A is wrong because AB sin θ gives the magnitude of the vector (cross) product, not the scalar product.
Why B is wrong: B is wrong because tan θ does not appear in either the dot or cross product definition.
Why D is wrong: D is wrong because AB / cos θ has no standard vector-product definition. The dot product multiplies by cos θ, not divides.
The vector product A × B is:
Answer: B. The cross product A × B = AB sin θ n̂, which is a vector perpendicular to the plane containing A and B, with direction given by the right-hand rule (NCERT Class 11 Physics Chapter 3, page 27).
Why A is wrong: A is wrong because AB cos θ defines the scalar (dot) product, not the cross product. The cross product is a vector, not a scalar.
Why C is wrong: C is wrong because the cross product is perpendicular to the plane of the two vectors, not lying within it.
Why D is wrong: D is wrong because while AB sin θ gives the correct magnitude, the cross product is a vector (with direction from the right-hand rule), not a scalar.
If î, ĵ, k̂ are unit vectors along the x, y, z axes, then î × ĵ equals:
Answer: B. By the right-hand rule and the standard cyclic relation for orthogonal unit vectors: î × ĵ = k̂ (NCERT Class 11 Physics Chapter 3, page 27).
Why A is wrong: A is wrong because î × ĵ = 0 would mean the unit vectors are parallel. They are perpendicular, so sin 90° = 1, giving a non-zero cross product.
Why C is wrong: C is wrong because the cross product of two vectors is a vector, not a scalar. The magnitude is 1, but the result is the vector k̂.
Why D is wrong: D is wrong because ĵ × î = −k̂ (anti-commutativity), but the question asks for î × ĵ, which is +k̂.
Two vectors have magnitudes 5 and 12, and the angle between them is 60°. Their scalar product is:
Answer: A. A · B = AB cos θ = 5 × 12 × cos 60° = 5 × 12 × 0.5 = 30. (Direct application of dot product definition.)
Why B is wrong: B is wrong because 60 = 5 × 12 without the cos 60° factor. Forgetting to apply cos θ is a common error.
Why C is wrong: C is wrong because 30√3 = 5 × 12 × (√3/2) = 5 × 12 × sin 60°. This is the magnitude of the cross product, not the dot product. The trap is substituting sin for cos.
Why D is wrong: D is wrong because 52 does not correspond to any correct application of the dot or cross product formula with these values.
If A = 3î + 4ĵ and B = 2î − 5ĵ, then A · B is:
Answer: D. Using component form: A · B = (3)(2) + (4)(−5) = 6 − 20 = −14. The dot product in Cartesian components is the sum of products of corresponding components.
Why A is wrong: A is wrong because 26 results from adding the absolute values (6 + 20) instead of respecting the negative sign in the j-component product. Sign errors in component multiplication are a common trap.
Why B is wrong: B is wrong because +14 reverses the sign of the correct answer. The j-component product is (4)(−5) = −20, not +20.
Why C is wrong: C is wrong because −23 does not match any correct combination of component products. This is likely an arithmetic slip from misreading a coefficient.
The magnitude of A × B when |A| = 3, |B| = 4, and the angle between them is 30° is:
Answer: A. |A × B| = AB sin θ = 3 × 4 × sin 30° = 3 × 4 × 0.5 = 6.
Why B is wrong: B is wrong because 12 = 3 × 4 without the sin 30° factor. This error comes from forgetting the angular dependence in the cross product magnitude.
Why C is wrong: C is wrong because 6√3 = 3 × 4 × (√3/2) = 3 × 4 × cos 30°. This is actually the dot product of these vectors, not the cross product magnitude. The trap is using cos instead of sin.
Why D is wrong: D is wrong because 3 does not correspond to any correct application of the cross product formula with these values.
Two vectors P and Q satisfy P · Q = 0. Which of the following must be true?
Answer: C. P · Q = PQ cos θ = 0 means either cos θ = 0 (i.e., θ = 90°, the vectors are perpendicular) or at least one of P, Q is zero. Both conditions make the dot product vanish.
Why A is wrong: A is wrong because parallel vectors (θ = 0° or 180°) give cos θ = ±1, so P · Q = ±PQ, which is zero only if one vector has zero magnitude — not the general case for parallel vectors.
Why B is wrong: B is wrong because when P ⊥ Q (θ = 90°), |P × Q| = PQ sin 90° = PQ, which is non-zero (assuming non-zero magnitudes). The cross product is maximum when the dot product is zero.
Why D is wrong: D is wrong because P = −Q means they are anti-parallel (θ = 180°), giving P · Q = −P², which is non-zero unless P = 0.
If A × B = A × C and A ≠ 0, can we conclude B = C?
Answer: D. A × B = A × C implies A × (B − C) = 0. This means either B − C = 0 (i.e., B = C) or B − C is parallel to A (sin θ = 0). Unlike scalar algebra, you cannot "cancel" a vector from a cross product. This is a standard conceptual NEET trap about the properties of the cross product.
Why A is wrong: A is wrong because the cross product does not obey a cancellation law. A × (B − C) = 0 only tells us B − C is parallel to A (or zero), not necessarily that B = C.
Why B is wrong: B is wrong because the conclusion from A × (B − C) = 0 is that (B − C) is parallel to A, not that B and C are perpendicular to A. Perpendicularity would make the cross product non-zero, not zero.
Why C is wrong: C is wrong because the magnitude of A being 1 does not change the algebraic structure. The non-cancellation property holds regardless of |A|.
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Given
A = 2î + 3ĵ − k̂, B = î − 2ĵ + 3k̂.
Required
(a) Find A · B.
(b) Find A × B.
(c) Find the angle between A and B.
Concept
The dot product uses component-wise multiplication and summation: A · B = AₓBₓ + AᵧBᵧ + A_zB_z. The cross product uses the determinant of a 3×3 matrix with unit vectors in the top row. The angle between vectors is found from cos θ = (A · B) / (|A||B|).
Formula
• A · B = AₓBₓ + AᵧBᵧ + A_zB_z• A × B = (AᵧB_z − A_zBᵧ)î − (AₓB_z − A_zBₓ)ĵ + (AₓBᵧ − AᵧBₓ)k̂• cos θ = (A · B) / (|A||B|)
Substitution
(a) A · B = (2)(1) + (3)(−2) + (−1)(3)
(b) A × B:• î component: (3)(3) − (−1)(−2) = 9 − 2 = 7• ĵ component: −[(2)(3) − (−1)(1)] = −[6 − (−1)] = −[6 + 1] = −7• k̂ component: (2)(−2) − (3)(1) = −4 − 3 = −7
(c) |A| = √(4 + 9 + 1) = √14; |B| = √(1 + 4 + 9) = √14
Calculation
(a) A · B = 2 − 6 − 3 = −7
(b) A × B = 7î − 7ĵ − 7k̂
(c) cos θ = −7 / (√14 × √14) = −7/14 = −0.5
∴ θ = 120°
Note on exact values: The vector components (2, 3, −1, 1, −2, 3) are exact integers (counting numbers defining the vectors), so they do not limit significant figures. The answer θ = 120° is exact.
Final answer
(a) A · B = −7
(b) A × B = 7î − 7ĵ − 7k̂ (or equivalently 7(î − ĵ − k̂))
(c) θ = 120°
Common trap
The sign trap in the ĵ component of the cross product: the determinant expansion has a negative sign in front of the ĵ cofactor. Writing +7ĵ instead of −7ĵ is a high-frequency error. Mnemonics: the middle component always carries a minus sign in the determinant expansion.
Similar NEET-style question
If P = î − ĵ + 2k̂ and Q = 3î + 2ĵ − k̂, find the angle between P and Q. (Answer: cos θ = (3 − 2 − 2)/(√6 × √14) = −1/√84; θ = cos⁻¹(−1/√84) ≈ 96.3°.)
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The scalar product or dot product of two vectors A and B is A·B = AB cos θ (Eq. 5.1a), where θ is the angle between them; since A, B and cos θ are scalars, A·B is a scalar. Work is defined as the scalar product of force and displacement.
-- NCERT Class 11 Physics, Ch. 5, p. 72A vector product of two vectors a and b is a vector c whose magnitude is c = ab sin θ (θ the angle between a and b) and which is perpendicular to the plane containing a and b. Moment of a force and angular momentum are defined as vector products.
-- NCERT Class 11 Physics, Ch. 6, p. 102More in Kinematics: 14 exam traps and mistakes · 6 formulas · 10 question patterns from its other lessons.
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