Using the technique of point source atom interferometry (PSI), we characterize the sensitivity of a multi-axis gyroscope based on free-space Raman interrogation of a single source of cold atoms in a glass vacuum cell. The instrument simultaneously measures the acceleration in the direction of the Raman laser beams and the projection of the rotation vector onto the plane perpendicular to that direction. The sensitivities for the magnitude and direction of the rotation vector measurement are 0.033 • /s and 0.27 • with one second averaging time, respectively. The fractional acceleration sensitivity δg/g is 1.6 × 10 −5 / √ Hz. The sensitivity could be improved by increasing the Raman interrogation time, allowing the cold-atom cloud to expand further, correcting the fluctuations in the initial cloud shape, and reducing sources of technical noise. The PSI technique resolves a rotation vector in a plane by measuring a phase gradient. This two-dimensional rotation sensitivity may be specifically important for applications such as tracking the precession of a rotation vector and gyrocompassing.