Simulation · Physics · Class 11
The vector product in 3D
From the lesson Vector product of two vectors in System of Particles and Rotational Motion. Change the values and watch what happens.
The vector product in 3DPhysics · Class 11
The idea behind it
NCERT §6.5
- The vector (cross) product c = a × b has magnitude ab sin θ, where θ is the smaller angle between a and b, and points perpendicular to the plane of a and b.
- Direction by the right-hand rule: curl the fingers of the right hand from a to b through the smaller angle; the thumb gives c. A right-handed screw turned from a to b advances along c.
- The cross product is not commutative: b × a = −(a × b), same size, opposite direction. The scalar product is commutative.
- Under reflection every component changes sign, so a → −a and b → −b, but a × b is unchanged.
- Both products are distributive: a × (b + c) = a × b + a × c. Also a × a = 0.
- Unit vectors: i × i = j × j = k × k = 0; i × j = k, j × k = i, k × i = j, and the products in the reverse order are negative.
- In components, a × b = (a_y b_z − a_z b_y) i + (a_z b_x − a_x b_z) j + (a_x b_y − a_y b_x) k, which is the determinant with rows (i, j, k), (aₓ, a_y, a_z), (bₓ, b_y, b_z).
- For a = 3i − 4j + 5k and b = −2i + j − 3k: a · b = −25 and a × b = 7i − j − 5k, so b × a = −7i + j + 5k.