The scalar product of the vector a = + + k with a unit vector along the sum of the vectors b =2 +4 - k and c = +2 +3 k is equal to one. Then the value of ' ' is

Step-by-step solution

Given a = i + j + k , b = 2 i + 4 j - k , and c = i + 2 j + 3 k . Calculate the sum vector d = b + c = (2+1) i + (4+2) j + (3- ) k = 3 i + 6 j + (3- ) k . The unit vector along d is d = d | d | = 3 i + 6 j + (3- ) k 3^2 + 6^2 + (3- )^2 . The scalar product a d = 1 implies a d | d | = 1 , so a d = | d | . Calculate a d = (1)(3) + (1)(6) + (1)(3- ) = 3 + 6 + 3 - = 12 - . Calculate | d | = 9 + 36 + (3- )^2 = 45 + (3- )^2 . Equating the two: 12 - = 45 + 9 - 6 + ^2 = 54 - 6 + ^2 . Squaring both sides: (12 - )^2 = 54 - 6