What is the vector u if u*v=12, u*w=14, v=6i+3j, w=2i+j ?

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.