find the angle theta between the lines whose direction cosines are given by the equations 3l+m+5n=0 and 6mn-2nl+5lm=0.

Dear student,

The direction cosines of a vector are the cosines of the angles between the vector and the three coordinate axes.

If v is a vector

where is a basis. Then the direction cosines are

Note that

α² + β² + γ² = 1

and

(α, β, γ) is the Cartesian coordinates of the unit vector