ABCD forms a rectangle. With 3 points, A,B,C, find D.

[says]

Homework Statement

Given A = [2, 9, 8], B = [6, 4, −2] and C = [7, 15, 7], show that AB and AC are perpendicular, then find D so that ABCD forms a rectangle.

Homework Equations

Dot Product

The Attempt at a Solution

The vector AB = B - A = [4,-5,-10]
The vector AC = C - A = [5,6,-1]

AB⋅AC = 0 if they are perpendicular

(4*5) + (-5*6) + (-10*-1) = 20 - 30 + 10 = 0

AB and AC are perpendicular.

I'm not sure how to find the point D though. Seeing as it's a rectangle, the distance from CD = AB? And the distance from BD = AC? Can I just use Pythagoras to find the distance from A to D:

AD² = AB²+AC²

AD² = [4,-5,-10]² + [5,6,-1]²

AD² = [41,61,101]

I got to here and feel like I might have over-thought the problem a bit...

I've arranged the letters below to show how I'm setting up the points in a rectangle

BD
AC

 

[mfb]

You can work with distances, but there is a much easier approach. What do you know about vectors of opposite sides in a rectangle, e.g, DC and AB?

 

[says]

They have the same length.

 

[mfb]

You can make a stronger statement (which is true even in general parallelograms, and gave them their name).

 

[says]

They are parallel, so one will be a multiple of the other?

 

[mfb]

They are parallel and have the same length. What does that make together?

 

[says]

CD = AB = [4,-5,-10]

So if we start from C = [7, 15, 7] we just add the vector to that to get the coordinates of D?

D (coordinates) = C+ AB = [7, 15, 7] + [4,-5,-10] = [11, 10,-3]

 

[mfb]

Right.