Can an alien see Earth and Venus simultaneously with 5π/6 radian field of view?

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Homework Statement


From an alien spaceship Earth is in the direction (1, -8, -4) and Venus is in the direction (3, 12, 4). The average alien eye has a field of view of 5pi/6 radians. Without using arccos or calculating any exact angles, determine if the average one eyed alien on the spaceship could see Earth and Venus at the same time. You may find it useful to use root(3)/2 approx equal to 0.866

Homework Equations



cos(Θ) = u . v / ||u|| ||v||

The Attempt at a Solution



if i find the dot product of the two vectors, it gives me -109, indicating that the angle between the two vectors is obtuse, however this might still mean that it is greater than 5pi/6 radians.

and now I'm stuck. The above formula tells me that the cos of the angle between the two vectors is -4.95, and using exact angles (which I know I can't do) cos(5pi/6) is -root(3)/2, but I don't know how to use all of this to answer the question
 
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nothingsus said:

Homework Statement


From an alien spaceship Earth is in the direction (1, -8, -4) and Venus is in the direction (3, 12, 4). The average alien eye has a field of view of 5pi/6 radians. Without using arccos or calculating any exact angles, determine if the average one eyed alien on the spaceship could see Earth and Venus at the same time. You may find it useful to use root(3)/2 approx equal to 0.866


Homework Equations



cos(Θ) = u . v / ||u|| ||v||

The Attempt at a Solution



if i find the dot product of the two vectors, it gives me -109, indicating that the angle between the two vectors is obtuse, however this might still mean that it is greater than 5pi/6 radians.

and now I'm stuck. The above formula tells me that the cos of the angle between the two vectors is -4.95
? You know that's impossible, don't you? The cosine of any angle must lie between -1 and 1. You have an error somewhere.

, and using exact angles (which I know I can't do) cos(5pi/6) is -root(3)/2, but I don't know how to use all of this to answer the question