As this question is tagged as catholic. I think I'll approach the catholic view (which certainly relates to all Trinitarian churches) but I will also bring up some doctrine of the eastern orthodox churches.
As Caleb puts it, this question is related with the Biblical basis for the doctrine of Trinity. Byrd and Software monkey answered outstandingly. The answers should be reviewed prior to this answer. Certainly there is a similarity amongst the questions, but JustinY's question seems to be more concerned with the relation that exists between Father, Son and Holy Spirit.
The core of the question relies in the following:
- That they all may be one; as thou, Father, art in me, and I in thee
- That they also may be one in us
To answer the first point it is important to bring back the doctrine of Trinity that explains that there is One God in three persons(ὑποστάσεις), with the same substance(οὐσία). To explain the difference between hypostasis and ousia is clearly stated by Basil of Caesare:
[...] ousia has the same relation to hypostasis as the common has to
the particular. Every one of us both shares in existence by the
common term of ousia, and, by his properties, is such an one and such
an one. In the same manner...the term ousia is common, like
goodness, or Godhead, or any similar attribute, while hypostasis is
contemplated in the property of Fatherhood, Sonship or the power of
Sanctification. St. Basil, Basil: Letters and Select Works. Quoted in
The Trinitarian Theology of St. Basil of Caesare by John Behr
Father and Son are the same ousia. Jesus himself teaches that there is One God in Mark 12: 28-31.
The second part of the question is how are the faithful are one in Christ. The answer may be found in John 15:4-5:
Remain in me, as I also remain in you. No branch can bear fruit by
itself; it must remain in the vine. Neither can you bear fruit unless
you remain in me. “I am the vine; you are the branches. If you remain
in me and I in you, you will bear much fruit; apart from me you can do
And may be expanded by recalling that the church is the body of Christ (1 Corinthians 12: 12-31) and it is Spouse of Christ.
To support this point one may use the following verse:
Now you are the body of Christ, and each one of you is a part of it.
(NIV 1 Corinthians 12: 27)
and also de article 796 of the Catechism of Catholic Church, of which I extract one excellent quote of St. Agustine:
This is the whole Christ, head and body, one formed from many . . .
whether the head or members speak, it is Christ who speaks. He speaks
in his role as the head (ex persona capitis) and in his role as body
(ex persona corporis). What does this mean? "The two will become one
flesh. This is a great mystery, and I am applying it to Christ and the
Church."240 and the Lord himself says in the Gospel: "So they are no
longer two, but one flesh."241 They are, in fact, two different
persons, yet they are one in the conjugal union, . . . as head, he
calls himself the bridegroom, as body, he calls himself "bride". (St.
Agustine. Enarratio in Psalmum 74, 4)
In few words, it is explained that the Church and Christ are one in a similar way in which husband and wife are one flesh. Because the Church is the wife that Christ loved for whom he gave himself up (Ephesians 5:25).
Hope this answers your question.