Physics · System Of Particles And Rotational Motion · NEET
The dot product A · B = AB cosθ gives a SCALAR (just a number, no direction) — it is largest when the vectors point the same way (θ = 0). The cross product A × B = AB sinθ n̂ gives a VECTOR — it is largest when the vectors are perpendicular (θ = 90°). So dot uses cos and answers 'how much overlap', while cross uses sin and answers 'how much they turn around each other'. NEET work (W = F · d) is a dot product; NEET torque (τ = r × F) is a cross product.
Because turning has a direction. When you rotate one vector toward another, the sense of rotation (clockwise or anticlockwise) needs an axis to point along. The cross product's direction IS that axis — it comes out perpendicular to the plane of A and B. This is exactly why torque and angular momentum, which describe turning, are vectors defined by cross products.
It is perpendicular to the plane containing both A and B, given by the right-hand rule: point the fingers of your right hand along A, curl them toward B, and your thumb points along A × B. NCERT states it as a right-handed screw turned from A to B — the direction it advances is A × B. There are two perpendicular directions to a plane; the right-hand rule picks the correct one.
No. The cross product is NOT commutative: A × B = −(B × A). Both have the same magnitude (AB sinθ) and both are perpendicular to the plane, but they point in OPPOSITE directions. This is because the right-handed screw turns A→B one way and B→A the opposite way. (Compare: the dot product IS commutative, A · B = B · A.)
A × A = 0 (the zero vector). Any vector crossed with itself is zero because the angle θ between them is 0°, and sin 0° = 0. The same applies to parallel vectors (θ = 0°) and antiparallel vectors (θ = 180°, sin 180° = 0). So î × î = ĵ × ĵ = k̂ × k̂ = 0.
Write î, ĵ, k̂ in a cycle (i → j → k → i). Going FORWARD in the cycle gives a PLUS unit vector: î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ. Going BACKWARD gives a MINUS: ĵ × î = −k̂, k̂ × ĵ = −î, î × k̂ = −ĵ. Any unit vector crossed with itself is 0. This cyclic rule is the fastest way to compute cross products in NEET.
The torque about the origin when a force 3ĵ N acts on a particle whose position vector is 2k̂ m is:
The moment of the force F = 4î + 5ĵ + 6k̂ acting at the point (2, 0, −3), about the point (2, −2, −2), is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
The magnitude is |A × B| = AB sinθ, where A and B are the magnitudes and θ is the angle between the two vectors (0° ≤ θ ≤ 180°). As a full vector, A × B = AB sinθ n̂, where n̂ is the unit vector perpendicular to the plane of A and B, pointing in the right-hand-rule direction.
Geometrically, |A × B| equals the area of the parallelogram formed by A and B as its two sides. Physically, the cross product measures the turning effect one vector has around a point — which is why torque (τ = r × F) and angular momentum (L = r × p) are defined as cross products in NEET rotational motion.
It is NEITHER commutative nor associative. A × B = −(B × A) (anticommutative), and A × (B × C) is generally not equal to (A × B) × C. However, the cross product IS distributive over addition: A × (B + C) = A × B + A × C.
It is ZERO when the two vectors are parallel or antiparallel (θ = 0° or 180°, since sin = 0) — including any vector crossed with itself, A × A = 0. It is MAXIMUM (value AB) when the vectors are perpendicular (θ = 90°, sin = 1).
Three key NEET quantities are cross products: torque τ = r × F, angular momentum L = r × p, and the velocity of a rotating particle v = ω × r. Knowing the magnitude rule (AB sinθ) and the right-hand direction rule lets you solve torque and angular momentum problems quickly, so this is a high-yield foundation topic.
