In the Bible, the phrase, “in the name of,” answers the question: “By what authority are you doing these things, and who gave you this authority?” (Matthew 21:23).
If my boss orders me to do something and I do it, I am acting in my boss’ name. Here are a couple examples from the Old Testament that use the phrase “in the name of”:In Deut 18:22, it says, “when a prophet speaks in the name of the LORD, if the word does not come to pass or come true, that is a word which the LORD has not spoken.”
David sent some young men to say something to Nabal. In 1 Sam 25:9, it says, “When David’s young men came, they said all this to Nabal in the name of David; and then they waited.”
Anything done in the name of Jesus, is done in the name of the Father as well since Jesus came in his Father’s name (John 5:24). Therefore, the phrase “in the name of Jesus” means the same as “in the name of the Father, and of the Son (Jesus).” Further, since the Spirit was sent in Jesus’ name (John 14:26), if I am moved by the Spirit to do something (such as baptize), I do it (baptize) in the name of the Father, and of the Son, and of the Holy Spirit because the Spirit acts in Jesus’ name, and Jesus acts in the Father’s name.