The angle between the lines whose direction cosines are - 3 4 , 1 4 , - 3 2 and - 3 4 , 1 4 , 3 2 is

Step-by-step solution

Let the direction cosines of the first line be (l₁, m₁, n₁) and those of the second line be (l₂, m₂, n₂) , where l₁ = - 3 4 , m₁ = 1 4 , n₁ = - 3 2 , and l₂ = - 3 4 , m₂ = 1 4 , n₂ = 3 2 . Using the formula for the cosine of the angle between the lines, = l₁ l₂ + m₁ m₂ + n₁ n₂ . Substituting the values, = ( - 3 4 ) ( - 3 4 ) + ( 1 4 ) ( 1 4 ) + ( - 3 2 ) ( 3 2 ) . Simplifying each term: 3 16 + 1 16 - 3 4 = 4 16 - 12 16 = - 8 16 = - 1 2 . Thus, = - 1 2 , which corresponds to an angle = 120^ since 120^ = - 1 2 . The