math. vector units. exc.?

Question

Find the angles vector B makes with the positive axes when vector b =i hat 6 - j hat 2 + k hat 3?

Answer 1

Length of this vector is
sqrt(6^2 +2^2 + 3^2)
=7
Hence the direction cosines are
6/7, -2/7, 3/7 and so the inverse cosines of these three values are the three angles.

I get
arcos(6/7) = 31 degrees (after conversion from radians)
and the other two are
106.6 degrees and 64.6 degrees.

Answer 2

let's write b as <6,-2,3>. cosine of angle between vectors a and b is
a•b/ |a||b|.

positive x axis is <1,0,0>, so
cos α = <6,-2,3>•<1,0,0> / |<6,-2,3>||<1,0,0>|
cos α = 6/ (√49 √1)
cos α = 6/7
α = 31°

positive y axis is <0,1,0>, so
cos ß = -2/7
ß = 106.6°

positive z axis is <0,0,1>, so
cos Γ = 3/7
Γ = 64.6°