Let the vector u be ai + bj.
u*v = (ai + bj)(6i + 3j) = 6a + 3b = 12
u*w = (ai + bj)(2i + j) = 2a + b = 14
Solve the simultaneous equations
6a + 3b = 12 and 2a + b = 14
We see that the system has no solutions as 6a + 3b = 12
=> 2a + b = 6
but 2a + b = 14
Therefore it is not possible to find a vector u which satisfies the given constraints.
We'll write the vector u as:
u = x*i + y*j
We'll apply the definition of dot product.
u*v = (xi + yj)(6i+3j)
u*v = 6xi^2 + 3xij + 6yij + 3yj^2
since the product of vectors ij = 0 and i^2 = j^2 = 1
u*v = 6x + 3y
We'll substitute the information given in enunciation and we'll obtain the following system:
6x + 3y = 12
2x + y = 4
y = 4 - 2x (1)
w*v = (xi + yj)(2i+j)
u*w = 2x + y
2x + y = 14 => y = 14 - 2x (2)
We'll equate (2) and (1):
14 - 2x = 4 - 2x
14 = 4 impossible
Since we've get an impossible equality, the vector u does not exist in the given conditions.