This paper proposes a new direct solution to the perspective-three-point (P3P) problem based on an algebraic approach. The proposed method represents the rotation matrix as a function of distances from the camera center to three 3D points, then, finds the distances by utilizing the orthogonal constraints of the rotation matrix. The formulation can be simply written because it relies only on some simple concepts of linear algebra. According to synthetic data evaluations, the proposed method gives the second-best performance against the state-of-the-art methods on both numerical accuracy and computational efficiency. In particular, the proposed method is the fastest among the quartic-equation based solvers. Moreover, the experimental results imply that the P3P problem still has an arguable issue on numerical stability regarding a point distribution and a camera pose.