A million years ago, an alien species built a vertical tower on a horizontal plane. When they returned they discovered that the ground had tilted so that measurements of 3 points on the ground gave coordinates of (0, 0, 0), (2, 1, 0), and (0, 3, 1). By what angle does the tower now deviate from the vertical?

Question

amistre64 · Answer

|dw:1385564380013:dw|

seems like an accurate depiction, define 2 vectors and take the cross product to determine the "normal to the plane" the angle this makes with the 0,0,1 vector tells how far it has leaned

amistre64 · Answer

if they coords they give represent abc,  then bxc should give us a normal to compare with

amistre64 · Answer

since the z axis of 0,0,1  has a length of 1 ... 

$$cos\alpha=\frac{\vec n\cdot (0,0,1)}{|n|}$$
$$ cos\alpha=\frac{\vec n_{Z}}{|n|}$$

amistre64 · Answer

can you work that out?

anonymous · Answer

I got (1,2,6) as the normal, do I get the dot product of n and (0,0,1) and divide it by the length of the normal?

amistre64 · Answer

yes, which you will soon find out is just the z component of the unit normal

anonymous · Answer

1/7

amistre64 · Answer

6/sqrt(1+4+36)

6/sqrt(41)

anonymous · Answer

sorry I forgot to square..
do I divide 6/sqrt(41) by length of n?

amistre64 · Answer

no, the length of n is sqrt41

the z component of the normal ... if we dot (x,y,z) with (0,0,1) we get 0+0+z = z

the z component is 6, the length is sqrt41

anonymous · Answer

do  I use the cosine rule?

amistre64 · Answer

id hope so ... since cos(angle) = 6/sqrt(41),  arc it

anonymous · Answer

0.3567333885

amistre64 · Answer

you set to radians or degrees?

anonymous · Answer

radians, I'm looking for the answer in radians

anonymous · Answer

it is correct, Thank you vey much for your help

amistre64 · Answer

making me have to reset the mode on my ti83 ... i should have you flogged!! :)