# Math Help - LinAlg - just one more question!

1. ## LinAlg - just one more question!

sorry for starting two new threads... i don't know how to combine them!

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), (1, 2, 0), and (0, 1, 3). By what angle does the tower now deviate from the vertical?

So I know that you have to form a triangle thing between the two planes... but I don't know the equation of the vertical tower plane... and what to do after that!

thanks =)

2. Actually, we prefer if you post separate questions on separate threads.

Basically, this problem amounts to finding the angle between two vectors - one that represents your original tower and another one that represents the tower that is now tilted.

So, since the original tower was vertical to the original ground, it is parallel to the vector $\bold{u} = (0,0,1)$ (assuming the z component is 'up' and the xy plane forms the original ground).

Now, use your 3 points on your 'new' ground, form the 2 vectors and take its cross product to get soem vector $\bold{v}$. This will yield a vector perpendicular to your tilted ground and so, it is parallel to your tilted tower.

So the angle between $\bold{u}$ and $\bold{v}$ should also be the angle between the original and tilted tower.

Use this relationship to find your angle: $\cos \theta = \frac{\bold{a} \cdot \bold{b}}{|\bold{a}| |\bold{b}|}$