Answer:
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:
[tex]\vec{A} = <3,3,3>[/tex]
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
[tex]|A| = \sqrt{3^2 + 3^2 +3^2} = \sqrt{27} = 3\sqrt{3}[/tex]
So the unit vector in the direction of vector A is
[tex]\vec{a} = \frac{\vec{A}}{|A|} = <\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}>[/tex]
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
[tex] a \cdot b = |a||b|cos\theta[/tex]
where Θ is the angle between vector a and b
[tex] a \cdot b = <\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}> <0,0,1> = 0 + 0 + \frac{1}{\sqrt{3}} = \frac{1}{\sqrt{3}}[/tex]
[tex]|a||b|cos\theta = 1*1*cos\theta[/tex]
therefore [tex]cos\theta = \frac{1}{\sqrt{3}}[/tex]
[tex]\theta = cos^{-1}\frac{1}{\sqrt{3}} = 0.955 rad [/tex] or 54.7 degrees
