Q.72. Vectors

Question 72.
A vector a = (x,y,z) makes an obtuse angle with y axis, makes equal angles with vector b = (y,-2z,3x) and  vector c = (2z,3x,-y) and is perpendicular to  vector d = (1,-1,2). If  vector a = 2*√3, then find vector a.

Answer 72.
a = (x,y,z) makes an obtuse angle with y axis
=> y is negative ... ( 1 )

vector a makes equal angles with vectors b and c
=> (a.b) / lal*lbl = (a.c) / lal*lcl
=> a.b = a.c
[because lbl = lcl = √(9x^2 + y^2 + 4z^2) and so lal*lbl calcel out with lal*lcl]
=> (x,y,z).(y,-2z,3x) = (x,y,z).(2z,3x,-y)
=> xy - 2yz + 3zx = 2zx + 3xy - yz
=> z(x - y) = 2xy
=> x - y = 2xy/z ... ( 2 )

vector a is perpendicular to vector d
=> a.d = 0
=> (x,y,z).(1,-1,2) = 0
=> x - y + 2z = 0
=> x - y = - 2z ... ( 3 )

lal = 2*√3
=> x^2 + y^2 + z^2 = 12 ... ( 4 )

Multiplying eqns. ( 2 ) and ( 3 )
=> (x - y)^2 = - 4xy
=> (x + y)^2 = 0
=> x = - y

Plugging y = - x in ( 2 )
2x = 2xy/z
=> z = y

Plugging x = - y and z = y in ( 4 )
3y^2 = 12
=> y = - 2
[minus because of ( 1 ). We decided to use ( 4 ) to find y (and not x or z) so that we can assign proper sign using condition ( 1 ).]

=> x = - y = 2 and z = y = - 2
=> a = 2i - 2j - 2k.

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