Let ABCDEFGH be a cube of side length 5, as shown.
Let P and Q be points on line AB and line AE, respectively, such that AP = 2 and AQ = 1.
The plane through C, P, and Q intersects line DH at R. Find DR.

Let →C=⟨0,0,0⟩Let →P=⟨5,3,0⟩Let →Q=⟨5,5,1⟩Let DR=zLet →R=⟨0,5,z⟩
The plane through C, P, and Q:
→x=→C+s⋅(→P−→C)+t⋅(→Q−→C)|→x=→R→C=→0→R=→0+s⋅(→P−→0)+t⋅(→Q−→0)→R=s⋅→P+t⋅→Q|→P=(530)→Q=(551)→R=(05z)(05z)=s⋅(530)+t⋅(551)
(1)0=5s+5t0=s+ts=−t(2)5=3s+5t5=3(−t)+5t5=2tt=52(3)z=0⋅s+tz=tz=52
DR = 52
