Those mouth speaketh vanity - Vain things; things not real and true; falsehood; lies. See the notes at Psalm 24:4. The idea is, that what they said had no foundation in truth - no reality. Truth is solid and reliable; falsehood is unreliable and vain.
And their right hand is a right hand of falsehood - The meaning here seems to be that even under the solemnities of an oath, when they lifted up their hands to swear, when they solemnly appealed to God, there was no reliance to be placed on what they affirmed or promised. Oaths were taken by lifting up the right hand as toward God. See Genesis 14:22; Exodus 6:8 (Margin, and Hebrew); Deuteronomy 32:40.