Another purpose of the incarnation was to accomplish God's redemption for man (Rom. 8:3; 1 Pet. 1:18-20; Heb. 9:26, 28, 12; 2:14). God cannot apply His redemptive work to us without being one with us. He died a vicarious death for us so that His death could now become our death (Gal. 2:20a). The only way this could be realized is by mingling. God is mingled with us, so now He is one with us. When He died on the cross, we died there with Him. Without our union with Him and without being joined to Him, His substitutionary death could not be applied to us. When we become one with Christ, whatever He has accomplished as our Substitute becomes ours.
The incarnation was also for the purpose of carrying out God's salvation in man (1 Tim. 1:15). We need not only God's redemption but also God's salvation. Redemption mostly deals with the negative things, and salvation is mostly to supply us with the positive things. For God to be our salvation, He needs to be one with us. Since He is one with us, His death was a vicarious death for our redemption. Also, His being everything to us as our life and nature means that He is our salvation. In order to be our salvation, He has to become one with us.
God was incarnated to impart the divine life into man. First John 4:9 says that "God sent His only begotten Son into the world that we might have life and live through Him." God came to be a man so that we could have His divine life. If He had never been a man, He could not come into us and we could not have Him as our divine life. Incarnation was for the impartation of the divine life into us.