For two vectors A = ai + bj + ck and B = a'i + b'j + c'k, the dot product between the two is given by A`.`B = a*a' + b*b' + c*c' = |A||B|cos A, where A is the angle between them.
The two vectors we have are: A = ai + 6j + 8k and B = 3i + 9j + 2k
If the two vectors are parallel, the angle between them is 0. cos 0 = 1
This gives A`.`B = 3a + 6*9 + 8*2 = sqrt(a^2 + 6^2 + 8^2)*sqrt(3^2 + 9^2 + 2^2)
=> 3a + 54 + 16 = sqrt(a^2 + 100)*sqrt 94
=> 3a + 70 = sqrt(a^2 + 100)*sqrt 94
square both the sides
9a^2 + 4900 + 420a = 94(a^2 + 100)
=> 9a^2 + 4900 + 420a = 94a^2 + 9400
=> 85a^2 - 420a + 4500 = 0
The roots of the equation are complex. For no value of a are the vectors parallel.
This could have also been determined from the fact that if the vectors are parallel, the ratio of a/a' = b/b' = c/c'; which is not the case here.
The vectors are perpendicular when the angle between them is 90. cos 90 = 0
=> 3a + 54 + 16 = 0
=> a = -70/3
For no value of a are the vectors parallel. They are perpendicular for a = -70/3