Solution :
Given :
a = (1, 2, 3, 4) ,    b = ( 4, 3, 2, 1),    c = (1, 1, 1, 1)     ∈   
a). (a.2c)b + ||-3c||a
Now, 
(a.2c) = (1, 2, 3, 4). 2 (1, 1, 1, 1) 
          = (2 + 4 + 6 + 6)
          = 20
-3c = -3 (1, 1, 1, 1)
      = (-3, -3, -3, -3)
||-3c|| = 
         
        
         = 6
Therefore, 
 (a.2c)b + ||-3c||a = (20)(4, 3, 2, 1) + 6(1, 2, 3, 4)   
                           = (80, 60, 40, 20) + (6, 12, 18, 24)
                          = (86, 72, 58, 44)
b). two vectors  and
 and  are parallel to each other if they are scalar multiple of each other.
 are parallel to each other if they are scalar multiple of each other.
i.e.,  for the same scalar r.
   for the same scalar r.
Given  is parallel to
 is parallel to  , for the same scalar r, we have
, for the same scalar r, we have

 ......(1)
   ......(1)
Let  ......(2)
   ......(2) 
Now given  and
  and   are perpendicular vectors, that is dot product of
 are perpendicular vectors, that is dot product of  and
  and   is zero.
 is zero.

 .......(3)
  .......(3)
Also given the sum of  and
  and   is equal to
 is equal to  . So
. So 


∴  ....(4)
   ....(4)
Putting the values of  in (3),we get
 in (3),we get

So putting this value of r in (4), we get


These two vectors are perpendicular and satisfies the given condition.
c). Given terminal point is  is (-1, 1, 2, -2)
 is (-1, 1, 2, -2)
We know that, 
Position vector = terminal point - initial point
Initial point = terminal point - position point
                   = (-1, 1, 2, -2) - (1, 2, 3, 4)
                   = (-2, -1, -1, -6)
d). 
 Let us say a vector  is perpendicular to
  is perpendicular to 
Then, 
      
      
There are infinitely many vectors which satisfies this condition.
Let us choose arbitrary 
Therefore, 
                       = -3
The vector is (-1, 1, 2, -3) perpendicular to given 