Question 328183
<pre>
Since A is the point (2,1,4) and B is the point (1,4,3), the vector from A to B
is 

< 1-2, 4-1, 3-4 > or < -1, 3, -1 >

Since B is the point (1,4,3) and C is the point (1,0,2), the vector from B to C
is 

< 1-1, 0-4, 2-3 > or < 0, -4, -1 >

Since those vectors are adjacent sides of a parallelogram, the
area of the parallelogram is the magnitude of their cross
product. So we find their cross product:

{{{abs(matrix(3,3,
        i,j,k,
        -1,3,-1,
        0,-4,-1))}}} = (-3-4)<b><i>i</b></i> - (1-0)<b><i>j</b></i> + (4-0)<b><i>k</b></i> = -7<b><i>i</b></i> - <b><i>j</b></i> +4<b><i>k</b></i>

And the magnitude of their cross product is 

{{{sqrt((-7)^2+(-1)^2+(4)^2)}}}{{{""=""}}}{{{sqrt(49+1+16)}}}{{{""=""}}}{{{sqrt(66)}}}
      
(b)

Let D be the point (x<sub>1</sub>, y<sub>1</sub>, z<sub>1</sub>).  Since A is the point (2,1,4),
then the vector from A to D is < x<sub>1</sub>-2, y<sub>1</sub>-1, z<sub>1</sub>-4 >

Since C is the point (1,0,2) and B is the point (1,4,3),

the vector from B to C is < 1-1, 0-4, 2-3 > or < 0, -4, -1 >

Since opposite sides of a parallelogram are equal and parallel,

they must be the same vector.  Therefore we set the corresponding

components  of the vector from A to D,  < x<sub>1</sub>-2, y<sub>1</sub>-1, z<sub>1</sub>-4 > and the vector from B to C < 0,-4,-1 >

equal:

x<sub>1</sub>-2 = 0, y<sub>1</sub>-1 = -4, z<sub>1</sub>-4 = -1,

so

x<sub>1</sub>=2, y<sub>1</sub>=-3, z<sub>1</sub>=3

and the point D is (x<sub>1</sub>, y<sub>1</sub>, z<sub>1</sub>) = (2,-3,3)

Edwin</pre>