We study the geometry of flag manifolds under different embeddings into a product of Grassmannians. We show that differential geometric objects and operations -- tangent vector, metric, normal vector, exponential map, geodesic, parallel transport, gradient, Hessian, etc -- have closed-form analytic expressions that are computable with standard numerical linear algebra. Furthermore, we are able to derive a coordinate descent method in the flag manifold that performs well compared to other gradient descent methods.