# Math Help - is this correct?? - vectors

1. ## is this correct?? - vectors

ok the ques is -
The points A,B,C in R^3 are position vectors
a=(1,1,2) b=(2,-1,4) c=(2,1,5)
a) write down the vectors AB and AC
b) Hence or otherwise find the position vector of the point D such that ABCD is a parallelogram.

solution -
i solved for
a) AB=(1,-2,2) and AC=(1,0,3)
b) ok the answer differs from my answer but this is what i said -
using the definition of vector addition,
a+b= c
AB=-DC=-(1,-2,2)
AC=(1,0,3)
DC + AD = AC
-(1,-2,3) + (x,y,z) =(1,0,3)
therefore,
(x,y,z) = (2,-2,5)

am i incorrect for ques (b)??
the answer was different!

2. Let $d=(x,y,z)$

If M is the point of intersection of diagonals AC and BD, then M is the midpoint of AC and BD.

Then $x_M=\frac{x_A+x_C}{2}=\frac{x_B+x_D}{2}\Rightarrow x_D=1$

$y_M=\frac{y_A+y_C}{2}=\frac{y_B+y_D}{2}\Rightarrow y_D=2$

$z_M=\frac{z_A+z_C}{2}=\frac{z_B+z_D}{2}\Rightarrow z_D=3$

$\Rightarrow d=(1,2,3)$