Verify that the points are the vertices of a parallelogram. $A ( 2,2,2 ) , B ( 3,4,5 ) , C ( 7,6,3 ) , D ( 8,8,6 )$
A) $\overrightarrow { A B } = ( 1,2,3 )$ is parallel to $\overrightarrow { C D } = ( 1,2,3 )$
$\overrightarrow { B D } = ( 1,4,5 )$ is parallel to $\overrightarrow { A C } = ( 1,4,5 )$
Opposites are parallel and same length. So $A B C D$ form a parallelogram.
B) $\overrightarrow { A B } = ( 1,2,3 )$ is parallel to $\overrightarrow { C D } = ( 5,4,1 )$
$\overrightarrow { B D } = ( 5,4,1 )$ is parallel to $\overrightarrow { A C } = ( 1,2,3 )$
Opposites are parallel and same length. So $A B C D$ form a parallelogram.
C) $\overrightarrow { A B } = ( 1,2,3 )$ is not parallel to $\overrightarrow { C D } = ( 1,2,3 )$
$\overrightarrow { B D } = ( 5,4,1 )$ is not parallel to $\overrightarrow { A C } = ( 5,4,1 )$
Opposites are not parallel and same length. So $A B C D$ form a parallelogram.
D) $\overrightarrow { A B } = ( 1,2,3 )$ is perpendicular to $\overrightarrow { C D } = ( 1,2,3 )$
$\overrightarrow { B D } = ( 5,4,1 )$ is perpendicular to $\overrightarrow { A C } = ( 5,4,1 )$
Opposites are perpendicular and same length. So $A B C D$ form a parallelogram .
E) $\overrightarrow { A B } = ( 1,2,3 )$ is parallel to $\overrightarrow { C D } = ( 1,2,3 )$
$\overrightarrow { B D } = ( 5,4,1 )$ is parallel to $\overrightarrow { A C } = ( 5,4,1 )$
Opposites are parallel and same length. So $A B C D$ form a parallelogram.

Question

Verify that the points are the vertices of a parallelogram.  $A ( 2,2,2 ) , B ( 3,4,5 ) , C ( 7,6,3 ) , D ( 8,8,6 )$ 
A) $\overrightarrow { A B } = ( 1,2,3 )$ is parallel to $\overrightarrow { C D } = ( 1,2,3 )$
$\overrightarrow { B D } = ( 1,4,5 )$ is parallel to $\overrightarrow { A C } = ( 1,4,5 )$
Opposites are parallel and same length. So $A B C D$ form a parallelogram.
B) $\overrightarrow { A B } = ( 1,2,3 )$ is parallel to $\overrightarrow { C D } = ( 5,4,1 )$
$\overrightarrow { B D } = ( 5,4,1 )$ is parallel to $\overrightarrow { A C } = ( 1,2,3 )$
Opposites are parallel and same length. So $A B C D$ form a parallelogram.
C) $\overrightarrow { A B } = ( 1,2,3 )$ is not parallel to $\overrightarrow { C D } = ( 1,2,3 )$
 $\overrightarrow { B D } = ( 5,4,1 )$ is not parallel to $\overrightarrow { A C } = ( 5,4,1 )$
Opposites are not parallel and same length. So $A B C D$ form a parallelogram.
D) $\overrightarrow { A B } = ( 1,2,3 )$ is perpendicular to $\overrightarrow { C D } = ( 1,2,3 )$ 
$\overrightarrow { B D } = ( 5,4,1 )$ is perpendicular to $\overrightarrow { A C } = ( 5,4,1 )$
Opposites are perpendicular and same length. So $A B C D$ form a parallelogram .
E) $\overrightarrow { A B } = ( 1,2,3 )$ is parallel to $\overrightarrow { C D } = ( 1,2,3 )$
$\overrightarrow { B D } = ( 5,4,1 )$ is parallel to $\overrightarrow { A C } = ( 5,4,1 )$
Opposites are parallel and same length. So $A B C D$ form a parallelogram.

Quizplus · Accepted Answer

To verify if the points form a parallelogram, we need to check if opposite sides are parallel (and thus have the same direction vectors) and have the same length. Calculating the direction vectors for each pair of opposite sides:- $\overrightarrow{AB} = B - A = (3-2, 4-2, 5-2) = (1,2,3)$- $\overrightarrow{CD} = D - C = (8-7, 8-6, 6-3) = (1,2,3)$- $\overrightarrow{AC} = C - A = (7-2, 6-2, 3-2) = (5,4,1)$- $\overrightarrow{BD} = D - B = (8-3, 8-4, 6-5) = (5,4,1)$Since $\overrightarrow{AB}$ is parallel and equal in length to $\overrightarrow{CD}$, and $\overrightarrow{AC}$ is parallel and equal in length to $\overrightarrow{BD}$, the points form a parallelogram.

Verify That the Points Are the Vertices of a Parallelogram A(2,2,2),B(3,4,5),C(7,6,3),D(8,8,6)A ( 2,2,2 ) , B ( 3,4,5 ) , C ( 7,6,3 ) , D ( 8,8,6 )A(2,2,2),B(3,4,5),C(7,6,3),D(8,8,6)

Verify That the Points Are the Vertices of a Parallelogram $A ( 2,2,2 ) , B ( 3,4,5 ) , C ( 7,6,3 ) , D ( 8,8,6 )$