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It sounds like you're using C++ and looking for a vector in the parent frame with the same direction as the x axis of the child frame, or vector D in this picture.

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You've got the geometry_msgs/Pose expressed as a tf::Transform, that's great! The basis of the transform is the rotation matrix to go from the child frame to the parent frame. An x axis vector is (1,0,0). To transform an x axis vector from the child frame to the parent frame you can use operator*(Matrix3x3, Vector3) from the following link

tf::Transform my_pose = ... //You already have this
tf::Vector3 x_axis(1,0,0);
tf::Vector3 D = myPose.getBasis() * x_axis;

Make sure to do matrix * vector and not vector * matrix. The latter will return the parent x axis expressed in the child frame.