Motion in a Plane – Vectors: NEET MCQs with Answers

Explanation: Vectors are added by the triangle law: if two vectors are placed head to tail, the single vector drawn from the free tail to the free head is their resultant. Reading the figure, d is directed vertically upwards and e is directed horizontally to the left. Placing the tail of e at the head of d, the closing side of the triangle runs upwards and to the left — which is exactly the vector f, both in magnitude and in direction. Hence d + e = f. The other combinations fail: b and c, d and c, and b and e each produce a resultant whose direction does not match that of f, so none of those statements is correct.

Explanation: The scalar product is F·r = (2)(3) + (1)(2) + (−1)(−2) = 6 + 2 + 2 = 10. For the vector product, the î term is (1)(−2) − (−1)(2) = 0, the ĵ term is −[(2)(−2) − (−1)(3)] = 1, and the k̂ term is (2)(2) − (1)(3) = 1. So F × r = ĵ + k̂, of magnitude √2. The pair is therefore 10 and √2.

Explanation: When two adjacent SIDES are given, the area is the full magnitude of their cross product, with no factor of a half. The cross product is î[(4)(−1) − (5)(−1)] − ĵ[(3)(−1) − (5)(1)] + k̂[(3)(−1) − (4)(1)] = î + 8ĵ − 7k̂. Its magnitude is √(1 + 64 + 49) = √114.

Explanation: In a scalar triple product the positions of the dot and the cross may be interchanged: A · (B × A) = (A × B) · A Alternatively, note that B × A is by definition perpendicular to both B and A. Since the result of the cross product is perpendicular to A, its dot product with A must be zero: A · (B × A) = 0 More formally, A · (B × A) = (A × A) · B, and since A × A = 0, the whole expression vanishes. In general, a scalar triple product is zero whenever any two of its three vectors are the same (or coplanar).

Explanation: The cross product of any vector with the null vector is itself a null vector, A × 0 = 0. A cross product is always a vector, so the result is the zero vector and not the scalar zero.

Explanation: A vector quantity must possess both magnitude and direction and must obey the laws of vector addition. Speed is the magnitude of velocity; it tells us only how fast an object moves and gives no information about the direction of motion, so it is a scalar. Velocity (rate of change of displacement), torque (τ = r × F, the cross product of two vectors) and displacement (the shortest directed path from initial to final position) all have direction associated with them and are therefore vectors.

Explanation: A unit vector has magnitude 1, so (0.5)² + (0.8)² + c² = 1. Thus 0.25 + 0.64 + c² = 1, giving c² = 0.11 and c = √0.11.

Explanation: Perpendicular sum and difference give (A + B)·(A − B) = 0, so A² − B² = 0 and |A| = |B|. The forces must be equal in magnitude, but their directions may differ, so they need not be equal as vectors.

Explanation: AB sin θ = √3 AB cos θ gives tan θ = √3, so θ = 60°. Then |A + B| = √(A² + B² + 2AB cos 60°) = (A² + B² + AB)^½.

Explanation: Two vectors are perpendicular when their scalar (dot) product vanishes. Let a = 2î + 3ĵ + 8k̂ and b = −4î + 4ĵ + αk̂. a · b = (2)(−4) + (3)(4) + (8)(α) = −8 + 12 + 8α = 4 + 8α Setting a · b = 0: 4 + 8α = 0 ⇒ 8α = −4 ⇒ α = −1/2.

Explanation: Squaring both sides of the given condition: |A + B|² = |A − B|² |A|² + |B|² + 2A · B = |A|² + |B|² − 2A · B 4A · B = 0 |A||B| cos θ = 0 Since neither vector is null, cos θ = 0 ⇒ θ = π/2 = 90°. Useful companion results: if |A + B| = |A| + |B| then cos θ = 1 (θ = 0°), and if |A + B| = ||A| − |B|| then cos θ = −1 (θ = 180°).

Explanation: By definition, |A × B| = AB sin θ and A · B = AB cos θ. Substituting into the given condition: AB sin θ = √3 AB cos θ Dividing both sides by AB cos θ: tan θ = √3 θ = 60°. This is a standard result worth remembering: the cross product exceeds the dot product by a factor √3 precisely at 60°, and the two are equal at 45°.

Explanation: Let the two vectors be P and Q with an angle φ between them. |P + Q|² = P² + Q² + 2PQ cos φ |P − Q|² = P² + Q² − 2PQ cos φ Equating the two, since |P + Q| = |P − Q|: P² + Q² + 2PQ cos φ = P² + Q² − 2PQ cos φ 4PQ cos φ = 0 Since P ≠ 0 and Q ≠ 0, cos φ = 0 ⇒ φ = π/2 = 90°. The two vectors are therefore perpendicular — geometrically, the diagonals of the parallelogram they form are equal, which makes it a rectangle.

Explanation: |Δv| = √(30² + 40²) = 50 m/s (perpendicular); a = 50/10 = 5 m/s².

Explanation: Displacement = (13−2)î + (14−3)ĵ = 11î + 11ĵ; average velocity = displacement/time = (11/5)(î + ĵ).