Two Angles Imply a Third

Geometry Level 3

Vector v ⃗ \vec{v} makes an angle of 30 30 degrees with vector u ⃗ 1 = ( 1 , 0 , 0 ) \vec{u}_1 = (1,0,0) , and an angle of 80 80 degrees with vector u ⃗ 2 = ( 0 , 1 , 0 ) \vec{u}_2 = (0,1,0) .

If all components of v ⃗ \vec{v} are positive, what angle (in degrees) does v ⃗ \vec{v} make with vector u ⃗ 3 = ( 0 , 0 , 1 ) \vec{u}_3 = (0,0,1) ?

Note: The desired answer is in the range 0 < θ < 9 0 ∘ 0 < \theta < 90^\circ


The answer is 62.04.

Steven Chase by Steven Chase , 37, USA

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2 solutions

Jordi Curto
Dec 11, 2019

Consider spheric coordinates and |vector v| = 1 . Vector v = r , angle theta , angle fi with r = 1 Changing to cartesian coordinates x , y , z .

x = cos (theta) * cos ( fi ) = cos (30 º)

y = cos (theta) * sin ( fi ) = cos (80 º)

operating and considering sin^2 (fi) + cos^2(fi) = 1

cos^2 (theta) = cos^2 (30º) + cos^2 (80º) => we get theta

Answer for the problem is 90º - theta => 62,0385 º

3 Helpful 1 Interesting 1 Brilliant 0 Confused

If the vector v makes angles α , β , γ α,β,\gamma with the x , y , z x, y, z axes respectively, (the given vectors are in these directions) then cos ⁡ 2 α + cos ⁡ 2 β + cos ⁡ 2 γ = 1 \cos^2 α+\cos^2 β+\cos^2 \gamma=1 . Here α = 30 ° , β = 80 ° α=30\degree, β=80\degree . So γ = cos ⁡ − 1 1 − cos ⁡ 2 α − cos ⁡ 2 β = 62.0385 ° \gamma=\cos^{-1}\sqrt {1-\cos^2 α-\cos^2 β}=\boxed {62.0385\degree}

3 Helpful 1 Interesting 1 Brilliant 0 Confused

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