balmy charm
A cube has a corner in the point O=(-5,7,-2). The corners of the cube which is closest to O is:
A=(-3,10,4), B=(-2,λ,0), and C.
Find the parameter λ and decide the two possible coordinates for C.
My first thought was that the vector OA and OB are perpendicular to each other and that they both are perpendicular to OC, aka "normal vector"
So the dot product between OA and OB has to be 0.
So after i calculated OA and OB, I applied the cross product on them which would give me the normal vector.
Have I correctly found the normal vector? I know I haven't completed the exercise yet
So C should be OC+O right? so (6,2,-3)+(-5,7,-2)=(1,9,-5) which is one of the possible locations for C?
and the other one can be found by subtracting O-OC?
mystic mica
Is the z-coordinate of O 2 or -2? You wrote different things in the comment and the picture.
mystic mica
Ah, ok.
mystic mica
λ is correct.
As for the point C: we have two choices depending on the direction of the perpendicular vector.
Choice 1: C = O - OA⨯OB.
Choice 2: C = O - OB⨯OA.
mystic mica
Ah! Sorry, forgot about one thing: the length of the cross product is too large: the square of the side length.
So, the correct formulas would be O - (OA⨯OB)/|OA| and O - (OB⨯OA)/|OA|.
balmy charm
Just curious, why is C=O-(OA⨯OB)/|OA|? shouldn't C=O+OC just like OC=O-C and OA=A-O etc
mystic mica
Just curious, why is C=O-(OA⨯OB)/|OA|? shouldn't C=O+OC just like OC=O-C and OA=...
OC = C - O, so C = O + OC. And OC is defined as (OA⨯OB)/|OA| or (OB⨯OA)/|OA|.
balmy charm
Ah! Sorry, forgot about one thing: the length of the cross product is too large:...
Using the formulas you gave me i get these coordinates for C
Btw, the coordinates for C2 are the same that I got when I tried to complete it without your help, so it feels good that I was on the right track anyway 🙂
mystic mica
Using the formulas you gave me i get these coordinates for C
Hm, C1 doesn't look correct.
Ah, I see the problem. OA⨯OB isn't correct.
Also, you don't really need to separately calculate OB⨯OA, since OB⨯OA = -OA⨯OB.
balmy charm
I think I see it too, C1 should be (1,5,1) right?
