Calculating Vector Cosines: Understanding the Proper Method

[Ry122]

I don't understand how they got these values for the cosines of the vector.
My understanding is that the angle between the unit vector and the x,y and z axises are used to determine the cosine of the vector, so for example, the vector creates a 45 degree angle with the x-axis and so the vector cosine for that axis is cos(180-45), but this doesn't give me the correct answer, so can someone tell me where I'm going wrong?
(I got 45degrees because the dome on top of the hut is a square and half of a right angle is 45 degrees)

 

[uart]

so can someone tell me where I'm going wrong?

You're taking the ratio of the projection in the x-axis direction to the projection in the x,z plane.
Instead you should just to take the ratio of the projection in the x-axis direction to the total vector length (and similarly for the other two direction cosines).

For example the projection in the x-axis direction is 10m. The projection in the x,z plane is 10 \, \sqrt{2}. You seem to be using 10 / (10 \, \sqrt{2}) when you should be using 10/16.2.