Many tasks in natural language processing require the alignment of word embeddings. Embedding alignment relies on the geometric properties of the manifold of word vectors. This paper focuses on supervised linear alignment and studies the relationship between the shape of the target embedding. We assess the performance of aligned word vectors on semantic similarity tasks and find that the isotropy of the target embedding is critical to the alignment. Furthermore, aligning with an isotropic noise can deliver satisfactory results. We provide a theoretical framework and guarantees which aid in the understanding of empirical results.