Q.Define vector product. Explain the properties of a vector product with two examples.
You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.
Start your 14-day free trial to unlock the full solution →The vector (cross) product of two vectors A and B is A × B = |A||B| sin θ n̂ — a vector of magnitude |A||B| sin θ, directed perpendicular to the plane of A and B (right-hand rule). Key properties: it is anti-commutative and the product of a vector with itself is zero; torque and angular momentum are common physical examples.
Definition. The vector product (or cross product) of two vectors A and B, inclined to each other at angle θ, is defined as:
A × B = (|A||B| sin θ) n̂
where n̂ is a unit vector perpendicular to the plane containing A and B, with its direction given by the right-hand rule (curl the fingers of the right hand from A to B; the thumb points along n̂). The result of a vector product is itself always a vector.
Property 1 — Not commutative (anti-commutative). Unlike ordinary multiplication or the scalar (dot) product, the vector product does not commute:
A × B = −(B × A)
This is because reversing the order of A and B reverses the sense of rotation used in the right-hand rule, so the resulting vector points in the opposite direction, even though its magnitude |A||B| sin θ stays the same.
Property 2 — Cross product of a vector with itself (or parallel vectors) is zero. If A and B are parallel (or the same vector), θ = 0°, so sin θ = 0, giving:
A × A = 0, and more generally A × B = 0 whenever A and B are parallel or anti-parallel.
Example 1 — Torque. The torque (moment of force) produced by a force F acting at a position vector r from a pivot is:
…
Unlock everything free for 14 days
- Full step-by-step solutions
- Concept-first explanations
- Methods, shortcuts & mistakes
- PYQ mapping + timed mock tests
Full access for 14 days. No credit card required.