Question

A line makes angles of measure 45° and 60° with the positive directions of the Y and Z axes respectively. Find the angle made by the line with the positive direction of the X-axis. 

Answer 1

Let α, β, γ be the angles made by the line with positive direction of X, Y and Z axes, respectively. Given β = 45º and γ = 60º.

Now cos45° = `1/sqrt2`, cos60° = `1/2`

cos²45° = `1/2`, cos²60° = `1/4`

The sum of the squares of the direction cosines is always 1:

cos²α + cos²β + cos²γ = 1

cos²α + cos²45º + cos²60º = 1

`cos^2α + 1/2 + 1/4 = 1`

`cos^2α = 1 - 3/4`

`cos^2α = (4-3)/4`

`cos^2α = 1/4`

cos α = ± `1/2`

Now, find α

α = `cos^(-1) (1/2)` = 60°

α = `cos^(-1) (-1/2)` = 120°

There are two lines satisfying given conditions. Their direction angles are 45º, 60º, 60º and 45º, 60º, 120º.