Question about vectors and rectangles in R3

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SUMMARY

The discussion centers on solving a geometry problem involving a rectangle ABCD in R3 with given vertices A(-1,2,3), B(2,6,-9), and D(3,q,8). The main tasks are to determine the coordinates of vertex C and the angle between the diagonals of the rectangle. The user initially struggled with the problem but later resolved it by recognizing that the dot product of the sides must equal zero due to the rectangle's perpendicular nature. The final coordinates of point D were confirmed as (3, 14, 8).

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  • Understanding of vector operations, specifically dot and cross products.
  • Familiarity with 3D coordinate geometry.
  • Knowledge of properties of rectangles in three-dimensional space.
  • Ability to solve for unknown variables in geometric contexts.
NEXT STEPS
  • Study the properties of rectangles in R3, focusing on vector representation.
  • Learn about the application of the dot product in determining angles between vectors.
  • Explore the use of the cross product to find perpendicular vectors in three dimensions.
  • Practice solving similar geometry problems involving vertices and angles in 3D space.
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Students preparing for geometry tests, educators teaching vector mathematics, and anyone interested in applying vector operations in three-dimensional geometry.

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Hi all, I was doing some review questions for my vectors test tomorrow and I have no idea how to even start this one. I hope any of you can help me :biggrin:

Homework Statement


The rectangle ABCD has vertices at A(-1,2,3), B(2,6,-9) and D(3,q,8).
Note that the "q" in coordinate D is a variable

a. Determine the coordinates of the vertex C
This is the question I am stuck on, no idea how to start. This question is in the dot/cross product section of the textbook, so I'm assuming we have to use either one to solve this problem:confused:

b. Determine the angle between the two diagonals of this rectangle.
Need the vertex C to figure out the diagonals...

Homework Equations


if a= (a1, a2, a3), b= (b1, b2, b3),

a.b = a1*b1 + a2*b2 + a3*b3
a.b = |a|*|b|*cosσ, where σ is the angle between a and b

The Attempt at a Solution


I found coordinate D to be (3, 14, 8) and am trying to use the fact that in this case, a.b = 0 because it is a rectangle and every side is perpendicular.

Any help is appreciated, thanks!

EDIT: Solved, I was just over-thinking it. Thanks anyways
 
Last edited:
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A comment on your use of the word "coordinate."

D is a point, not a coordinate ("I found coordinate D ... "). What you did was to find the missing coordinate of point D.
 

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