The acute angle between the line r = ( i + 2 j + k ) + ( i + j + k ) and the plane r (2 i + p j + k ) = 8 is ⁻¹ ( 2 3 ) , then the value of p are...

Step-by-step solution

The direction vector of the line is b = i + j + k . The normal vector to the plane is n = 2 i + p j + k . The angle between a line and a plane is given by: = | b n | | b | | n | We are given = ⁻¹ ( 2 3 ) , so = 2 3 . Substituting the vectors into the formula: 2 3 = |(1)(2) + (1)(p) + (1)(1)| 1^2 + 1^2 + 1^2 2^2 + p^2 + 1^2 2 3 = |p + 3| 3 p^2 + 5 Squaring both sides: 2 9 = (p + 3)^2 3(p^2 + 5) 2 3 = p^2 + 6p + 9 p^2 + 5 Cross-multiplying yields: 2(p^2 + 5) = 3(p^2 + 6p + 9) 2p^2 + 10 = 3p^2 + 18p + 27 Rearranging t