Question

What is cross-product used for?

Answer 1

Hint:Cross multiplication between two vectors gives a vector product which is perpendicular to both the initial vectors and lies in a plane perpendicular to both of them. This is an important tool for the determination of various physical quantities such as torque and dipole moment. There are two kinds of multiplication between vectors, dot multiplication and cross multiplication. The product of a dot multiplication is a scalar quantity and the product of a cross multiplication is a vector quantity.

Complete step by step solution:
Cross product is a type of multiplication between two vectors \[\vec a\] and \[\vec b\] in a three-dimensional coordinate system. This multiplication is also known as vector multiplication. The resultant product is called a vector product as it is also a vector perpendicular to both vectors \[\vec a\] and \[\vec b\].

Vector multiplication between two vectors is represented as \[\vec a \times \vec b\].
\[\vec a \times \vec b = \left| {\vec a} \right|\left| {\vec b} \right|\sin \theta \hat n\] ---- (1)
Where \[\theta \] is the angle between the two vectors \[\vec a\] and \[\vec b\], \[\left| {\vec a} \right|\left| {\vec b} \right|\sin \theta \] denotes the magnitude of the resultant vector and \[\hat n\]is a unit vector that denotes the direction of the resultant vector.

Vector multiplication is non commutative i.e. \[\vec a \times\vec b \ne \vec b \times \vec a\]
but \[\vec a \times \vec b = - (\vec b \times \vec a)\]--- (2)
Consider the two vectors \[\vec a\] and \[\vec b\],
\[\vec a = {a_1}\hat i + {a_2}\hat j + {a_3}\hat k\]---- (3)
\[\vec b = {b_1}\hat i + {b_2}\hat j + {b_3}\hat k\]----(4)
Here, \[\hat i\], \[\hat j\] and \[\hat k\] are the basis vectors corresponding to x, y and z-axis of the cartesian coordinate system. The cross product is calculated using the right-hand screw rule.

Using that, we have, \[\hat i \times \hat j = \hat k\]--- (5), \[\hat j \times \hat k = \hat i\]--- (6), \[\hat k \times \hat i = \hat j\]--- (7) and the reverse as per equation (2). Also, \[\hat i \times \hat i = \hat j \times \hat j = \hat k \times \hat k = 0\]
So, \[\vec a \times \vec b = ({a_1}\hat i + {a_2}\hat j + {a_3}\hat k) \times ({b_1}\hat i + {b_2}\hat j + {b_3}\hat k)\]= \[({a_2}{b_3} - {a_3}{b_2})\hat i + ({a_3}{b_1} - {a_1}{b_3})\hat j + ({a_1}{b_2} - {a_2}{b_1})\hat k\]
This product can also be calculated as the determinant of the following matrix:
\[\vec a \times \vec b = \left( {\begin{array}{*{20}{c}}{\hat i}&{\hat j}&{\hat k}\\{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\end{array}} \right)\]
Cross product is used in rotational mechanics to determine torque and angular momentum.
It is also used to determine the angle between two vectors and a resultant vector which is orthogonal to both.

Note: Cross product of two vectors is not commutative and zero when they are either parallel or coincide. The magnitude of the resultant vector is maximum when the two vectors are perpendicular to each other. It always follows the right-hand screw rule and cross multiplication in a direction opposite to that will result in a negative sign.