If the points A(3,0,p),B(-1,q,3) and C(-3,3,0) are collinear, then find
(1) the ratio in which the points C divides the line segment AB
(2) the value of p and q.

Explanation

Let `bar(a),bar(b)` and bar(c) be the position vectors of A, B and C respectively.
Then `bar(a)=3hat(i)+0*hat(j)+phat(k),bar(b)=-hat(i)+qhat(j)+3hat(k)` and `bar(c)=-3hat(i)+0*hat(k)`.
(1) As the points A,B,C are collinear, suppose the points C divides line segment AB in the ratio `lamda:1`.
`:.` by the section formula,
`bar(c)=(lamda*bar(b)+1*bar(a))/(lamda+1)`
`:.-3hat(i)+3hat(j)+0*hat(k)=(lamda(-hat(i)+qhat(j)+3hat(k))+(3hat(i)+0*hat(j)+phat(k)))/(lamda+1)`
`:.(lamda+1)(-3hat(i)+3hat(j)+0*hat(k))=(-lamdahat(i)+lamdaqhat(j)+3lamdahat(k))+(3hat(i)+0*hat(j)+phat(k))`
`:.-3(lamda+1)hat(i)+3(lamda+1)hat(j)+0*hat(k)=(-lamda+3)hat(i)+lamdaqhat(j)+(3lamda+p)hat(k)`
By equality of vectors, we have,
`-3(lamda+1)=-lamda+3` . . . (1)
`3(lamda+1)=lamdaq` . . . (2)
`0=3lamda+p` . . . (3)
From equation (1) `-3lamda-3=-lamda+3`
`:.-2lamda=6" ":.lamda=-3`
`:.` C divides segment AB externally in the ratio 3:1.
(2) Putting `lmda=-3` in equation (2) we get,
`3(-3+1)=-3q`
`:." "-6=-3q" ":.q=2`
Also, putting `lamda=-3` in equation (3), we get,
`0=-9+p" ":.p=9`
Hence `p=9` and `q=2`.