Question

There are 4 points in space: A(0,0,1), B(2,-2,0), C(5,2,-12), and D(3,4,-11). Is ABCD a parallelogram? Justify your answer.

Answer 1

I have figured out the length of each side and graphed it. I am wondering how else I could show this is a parallelogram... ie.. how can I show that the sides are parallel... that the increase is the same along the 2 axis?

Answer 2

ok... but u need to find out a pair of sides equal in length....

then consider the coordinates of the 2 points in a one sides
A = (x1 , y1 ) B =( x2 , y2)
then the slope of the line is ---- ( y2 - y1 ) / (x2 - x1)

then consider the 2 pairs of coordinates in the other side ....and do the same and find the slope..
the 2 slopes should be equal ....

the u can say it;s a parallelogram since a pair of sides r equal and parallael

Answer 3

got it ?

Answer 4

got it. thanks, but how do i deal with slope in 3 dimensions? I plotted the points and labeled them so I could see what the parallel sides would be. I can also see from the plots that the slope between the x, y, and z coordinate are the same. How do I deal with the 3d slope mathematically?

Answer 5

mmmm....srry.. but i've got no idea about 3d slopes... perhaps u should ask it from @ganeshie8 ... he could help u :)

Answer 6

alright, thanks.

Answer 7

u r welcome...

Answer 8

opposite sides parallel <=> parallelogram

Answer 9

we may simply show that the opposite sides have same direction ratios and conclude right ?

Answer 10

what are direction ratios?

Answer 11

or rather.. how do you calculate those.

Answer 12

heard of direction-vector before ?

Answer 13

ahh yes, but this is for an assignment before the vectors section.

Answer 14

oh ok...

Answer 15

is there a way to show without the use of vectors?

Answer 16

yeah linear algebra

Answer 17

just show that, two points depend on two other points, th

Answer 18

I can see the slopes between the points and paralell sides are the same.. both graphically and by just counting. Calc III class so no Linear as of yet.

Answer 19

oh yes,
oly slopes is sufficient to conclude given points make a parallelogram. no need to find distancces ?

Answer 20

what are the least number of things u need to know if its a paralleogram

Answer 21

opposite sides parallel <=> its a parallelogram

Answer 22

so short of vectors and linear algebra.. for me to prove that the sides are parallel, I would ...draw the graph and show that the slope of the lines are the same on opposite sides? can i do that using the individual coordinates?

Answer 23

AD is the exact same distance between the points as BC.

Answer 24

bad drawing.. but