A straight line passes through the point (1,1,1) makes an angle 60^ with the positive direction of z -axis, and the cosine of the angles made by it with the positive directions of the y -axis and the x -axis are in the ratio 3 : 1 . What is the acute angle between the two possible positions of the line?

Step-by-step solution

given angle from z -axis =60^ n= 60^ = 1 2 Let the cosines of angle made by line with y -axis and aligned & x -axis be 3 x and x & l^2+m^2+n^2=1 & 3 x^2+x^2+ 1 4 =1 & x= 3 4 & 3 x= 3 4 & x= 3 4 aligned aligned & (l, m, n)= ( 3 4 , 3 4 , 1 2 ) & or ( -3 4 , - 3 4 , 1 2 ) & for angle & |l₁ l₂+m₁ m₂+n₁ n₂ | & = | -9 16 - 3 16 + 1 4 | & = | -1 2 |= & Angle = =60^ aligned