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

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### 2 Answers

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.**