Help me solve this question.. thx?

Question

The position vectors of two points A and B relative to an origin O are 3p-q and 2q-p respectively. Find, in terms of p and q, the position vector of the point C such that AC= 3BC.(AC and BC with the --> arrow on top u know what i mean)

Answer 1

|AC| = (x - 3)^2 + (y + 1)^2 = 3[(x + p)^2
x - 3 = 3(x + 1)
x - 3 = 3x + 3
2x = - 6
x = - 3p
y + 1 = 3(y - 2)
y + 1 = 3y - 6
2y = 7
y = (7/2)q
C = (7/2)q - 3p

check:
(-6)^2 + (9/2)^2 =? 9[(-2)^2 + (3/2)^2]
36 + 81/4 = 9(4 + 9/4)

Answer 2

I will try to solve your problem without using arrow heads
--> due to writing difficulties. The order of letters will indicate the direction of the vectors.

Now since AC = 3BC, we can write the same, by introducing

the origin O, as:

OC - OA = 3(OC - OB)

or OC - 3OC = OA - 3OB

or - 2OC = OA - 3OB

or OC = ½ (3OB - OA) = ½ [3(2q - p) - (3p - q)]

= ½ [6q - 3p - 3p + q]

= ½ (7q - 6p).