Contour location$\unicode{x2014}$the process of sequentially training a surrogate model to identify the design inputs that result in a pre-specified response value from a single computer experiment$\unicode{x2014}$is a well-studied active learning problem. Here, we tackle a related but distinct problem: identifying the input configuration that returns pre-specified values of multiple independent computer experiments simultaneously. Motivated by computer experiments of the rotational torques acting upon a vehicle in flight, we aim to identify stable flight conditions which result in zero torque forces. We propose a "joint contour location" (jCL) scheme that strikes a strategic balance between exploring the multiple response surfaces while exploiting learning of the intersecting contours. We employ both shallow and deep Gaussian process surrogates, but our jCL procedure is applicable to any surrogate that can provide posterior predictive distributions. Our jCL designs significantly outperform existing (single response) CL strategies, enabling us to efficiently locate the joint contour of our motivating computer experiments.