A cube is 3 cm on a side, with one corner at the origin. What is the unit vector pointing from the origin to the diagonally opposite corner at location <3,3,3> cm? what is the angle from this disagonal to one of the adjacent edges of the cubes?

0.955 rad or 54.7 degrees

Explanation:

The vector (let's call it's A) pointing from the origin to the diagonally opposite corner would have its value of:

but unit vector only has a length of 1, so we need to divide this vector by length of itself to it has the same direction but magnitude of 1

So the unit vector in the direction of vector A is

Let another unit vector b <0,0,1> lies in the adjacent edges of the cubes. This unit vector also has a length of 1. So |b| = 1.The dot product between a and b yields

where Θ is the angle between vector a and b

therefore

or 54.7 degrees