You asked since God is Triune why doesn't man made in His image also not have three parts? He does. Man is spirit, soul and body according Heb. 4.12 and 1 Thess. 5.23. For a deep study on the "dividing of spirit, soul and body" (Heb. 4.12) study the foremost book on the dividing...

http://www3.telus.net/trbrooks/SMCFP.htm

Your body provides you world-consciousness, your soul gives you self-consciousness, and you spirit gives you God-consciousness.

Your spirit though we don't know what it is made of, we do know it has 3 functions: intuition, communion and conscience. Your conscience operates according to intuition, and a person who is saved can commune with God through his intuitive spirit. Those who do not want a quickened conscience in Christ freely choose Hell.

Your soul has the functions of mind, will and emotion. Your mind, of course, can reason and imagine. Your will is the center of your very person that defines you by your choices or volition. And your emotion has 3 functions: affection (from love to hate), desire, and sensing/feeling. Emotion, ultimately, is the expressing of your feelings.

And your body, of course, has its 5 senses it makes contact the world with.

While God is 1/3 1st Person, 1/3 2nd Person, 1/3 3rd Person (total equaling 1) and each Person (not referring to three humans, but personal expressions in His 3 parts) is God (but the 3 Persons never overlap) even before He created time and space, man however is not 3 Persons for only God can be 3 Persons. This is what makes God unique and complicated to the point you can't fully understand Him. After all how do you fully understand that which is infinitely greater than you?

The Trinity has 3 independent wills of 3 Persons. Man has three minds (similar to but not quite like the 3 Persons of the Godhead); that is to say, the mind of the body, mind of the soul and mind of the spirit. These three pulls, if you will, seem to have a mind of their own. And this is as close as you can get to be constructed in God's image. What God is doing through the redemptive design, the 66 books of the Bible, is to get you to walk by the spirit and secondarily by your soul and body.

Your soul should be like the steward who receives instructions from the mistress (spirit) and then the servant (body) performs those actions according to the steward's command. It is like light, light bulb and wire. The electricity is conducted along the wire (spirit) and enters the bulb (soul) that gives us light and enlivens us (body). So it was in the Temple of ancient Israel. The Holiest of Holies where no man dare enter is our spirit. Only God can. The Holy Place outside the curtain where the priests can enter is our soul where many soulical functions take place as depicted by the two loafs of bread (free will, choice), lampstand with 7 candles (7 emotional areas of our life of the 7 feasts), incense (prayers taken up, mind). And the outer court where the sacrifice takes place on the altar for all to see is our body. Inside the Holiest of Holies is another revelation that of the Ark of the Covenant in which God sits on the mercy seat. Inside the Ark are 3 items: rod that buds, 10 commandments, and manna from God. The fig branch is like the Holy Spirit that buds in our spirit's intuition for those of us who are born-again. The manna is the word of God (66 books of the Bible) for our soul to read daily and feed on in communion (food for our spirit). And the 10 commandments are God's law that speak to our conscience which judge us silently independent of our soulical persuasions.

Praise the Lord! Amen.