The acute angle between the vector 2 i + j - 3 k and the plane containing the vectors 2 i + 3 j - k and i - j + 2 k is

Step-by-step solution

Let v = 2 i + j - 3 k . The normal to the plane containing a = 2 i + 3 j - k and b = i - j + 2 k is given by n = a b . n = (2 i + 3 j - k ) ( i - j + 2 k ) = 5 i - 5 j - 5 k A vector parallel to the normal is n ' = i - j - k . The angle between the vector v and the plane is given by = | v n '| | v | | n '| . v n ' = (2)(1) + (1)(-1) + (-3)(-1) = 4 | v | = 2^2 + 1^2 + (-3)^2 = 14 | n '| = 1^2 + (-1)^2 + (-1)^2 = 3 = 4 14 3 = 4 42 = ⁻¹ ( 4 42 ) Answer: ⁻¹ ( 4 42 )