Q.Define cross-product of two vectors. Give its geometrical meaning. Discuss some of its important properties.

The cross product of two vectors A and B is A x B = |A| |B| sin(theta) n-hat, a vector perpendicular to both, whose magnitude equals the area of the parallelogram they span.

Definition: The cross product (vector product) of two vectors A and B, inclined at angle theta to each other, is defined as:

A x B = |A| |B| sin(theta) n-hat

where |A| and |B| are the magnitudes of the vectors, theta is the angle between them (0 <= theta <= 180 degrees), and n-hat 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 n-hat).

Geometrical meaning: The magnitude |A x B| = |A||B| sin(theta) is numerically equal to the area of the parallelogram that has A and B as its two adjacent sides. This is because the parallelogram's area is base x height = |A| x (|B| sin theta), which is exactly the cross product's magnitude.

Important properties:

1. Not commutative: A x B = -(B x A) (reversing the order reverses the direction, since the right-hand rule flips).
2. Distributive over addition: A x (B + C) = A x B + A x C.
3. Cross product of a vector with itself is zero: A x A = 0 (since theta = 0, sin 0 = 0).
4. Magnitude is maximum when A and B are perpendicular (theta = 90 degrees), and zero when they are parallel or antiparallel (theta = 0 or 180 degrees).
5. For the standard unit vectors: i x j = k, j x k = i, k x i = j (and i x i = j x j = k x k = 0).
6. The cross product is used to define quantities such as torque (tau = r x F) and angular momentum (L = r x p). …