What should be the value of a if A = ai + 6j + 8k and B = 3i + 9j + 2k are parallel vectors and if they are perpendicular vectors.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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

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