 Hello and welcome to this session. In this session we are going to discuss Lagrange's identity. Lagrange's identity states that if a, b are any vectors then vector a cross vector b be whole square to be equal to modulus of vector a square into modulus of vector b square minus vector a dot vector b be whole square which can also be written in the determinant form as the determinant containing elements vector a dot vector a, vector a dot vector b, vector a dot vector b, vector b dot vector b let m cap be a unit vector dependent to the plane of vector a and vector b for right handed rotation from vector a to vector b, unit vector m cap is possible let theta be the angle between vector a and vector b then vector a cross vector b is equal to modulus of vector a into modulus of vector b into sin of angle theta into m cap which is equal to a into b into sin of angle theta into m cap where modulus of vector a is equal to a and modulus of vector b is equal to b now vector a cross vector b be whole square is equal to a square into b square into sin square theta into m cap square which is equal to a square into b square into sin square theta into 1 that is a square into b square into sin square theta as m cap square is equal to 1 now a square into b square into sin square theta can be written as a square into b square into 1 minus cross square theta that is a square into b square into 1 is a square into b square minus of a square into b square into cos square theta and which can be written as modulus of vector a square into modulus of vector b square minus of vector a dot vector b b a square as we know that vector a dot vector b is equal to modulus of vector a into modulus of vector b into cos of angle theta which can be written as a into b into cos of angle theta which implies that vector a cross vector b be whole square or modulus of vector a cross vector b be whole square is equal to vector a dot vector a into vector b dot vector b minus of vector a dot vector b into vector a dot vector b which can also be written in the determinant form as the determinant containing elements vector a dot vector a vector a dot vector b vector a dot vector b vector b dot vector b where a is equal to modulus of vector a b is equal to modulus of vector b and theta which lies between 0 and 5 is the angle between vector a and vector b this complete high session hope you enjoyed this session