Bra-ket notation can be effectively represented in LaTeX using the commands \langle and \rangle for the brackets. The discussion revolves around finding the vectors |v> and |w> in both complex and real two-dimensional vector spaces given specific inner product values. In the complex case, the inner products are <v|v>=2, <w|w>=4, and <v|w>=i. For the real case, the inner products are <v|v>=2, <w|w>=4, and <v|w>=1. The conversation emphasizes the need to derive the vectors based on these inner product conditions, with a focus on understanding the implications of the given values in both contexts before moving on to the complex scenario.