The angle between the line r = ( i + 2 j + k ) + ( i + j + k ) and the plane r (2 i - j + k ) = 8 is

Step-by-step solution

The direction vector of the given line is b = i + j + k . The normal vector to the given plane is n = 2 i - j + k . The angle between a line and a plane is given by the formula: = | b n | | b | | n | Calculating the dot product: b n = (1)(2) + (1)(-1) + (1)(1) = 2 - 1 + 1 = 2 Calculating the magnitudes of the vectors: | b | = 1^2 + 1^2 + 1^2 = 3 | n | = 2^2 + (-1)^2 + 1^2 = 4 + 1 + 1 = 6 Substituting these values into the formula: = 2 3 6 = 2 18 = 2 3 2 = 2 3 = ⁻¹ ( 2 3 ) Answer: = ⁻¹ ( 2 3 )