The angle between two nontrivial linear manifolds in the high dimensional Euclidean space can be characterized as a quadratic programming with quadratic equation constraints (QP-QEC). The computing of such angles has many important applications in statistics and data analysis, such as the canonical correlation analysis and the kernel correlation analysis between two multivariate random vectors. This paper explores the relationship between a more general QP-QEC and its corresponding generalized eigenvalue problem in terms of the KKT conditions. On this basis, we design an algorithm for computing such an angle by means of the solution method of the generalized eigenvalue problem.