A single GNSS record would give you the absolute position, not the attitude. For the attitude, you would need a second position after displacement and then measure the displacement vector. The longer the distance, the more precise estimation you will get, but this assumes that your attitude is constant during the displacement. For instance, with a good DGPS and 3 cm precision, an attitude based on 3 cm displacement could be completely unreliable, but it would be quite good with 10 m displacement. For static attitude you should look for INS systems
The attitude determination of the moving vehicle via GPS is possible only with multi-antenna and multi-receiver GPS systems. For the practical purposes it is necessay to have 3 such systems [1,2]. The accuracy of the attitude determination, which could be achieved by these systems, strictly depends on the distances between these three antennas. The estmations of this accuracy you can find in the following references:
1. Lachapelle G. Navigation Accuracy for Absolute Positioning // Lecture series: AGARD-LS-207, NATO.– 1996. – pp. 4–1, 4 – 10.
2. Lachapelle G. Attitude Determination // Lecture series: AGARD-LS-207,”System Implications and Innovative Applications of Satellite Navigation”, NATO.– Kiev, 1997. – pp. 10–1, 10 – 11.
I am sure that you can find more updated publications of Prof. Gerard Lachapelle (University of Calgary, Canada) devoted to this topic.