The posture Vivekananda prescribes, holding "the chest, neck, and head — in a straight line", is not his own formula. It is the Śvetāśvatara Upanishad's: Max Müller's translation in the reader (Sacred Books of the East 15, second Adhyāya, verse 8) has the wise man hold his body with its three erect parts, chest, neck and head, even. Vivekananda cites Śaṅkara's commentary on the same Upanishad a page later for the breathing exercise, but leaves this borrowing unmarked. It matters because the lectures present Raja Yoga as a science to be tested by experiment; much of their practical instruction is old scripture, given without its address.
echoes The Upanishads, Part II — SBE Vol. 15 — trans. F. Max Müller (1884)