# What is vector b if a*b=14 and c*b=16? a=3i+j, c=5i-2j

### 2 Answers | Add Yours

We'll write the vector b as:

b = x*i + y*j

We'll apply the definition of dot product.

a*b = (3i+j)(xi + yj)

a*b = 3xi^2 + 3yij + xij + yj^2

since the product of vectors ij = 0 and i^2 = j^2 = 1

a*b = 3x + y

c*b = 5x - 2y

We'll substitute the information given in enunciation and we'll obtain the following system:

3x + y = 14 =>y=14-3x

5x - 2y = 16=> 5x - 2 (14-3x) = 16=> 5x - 28 + 6x - 16 =0

11x = 44 =>x=4

y = 14 - 3*4

y = 14-12

y=2

So, the vector we are looking for is b**=4*i + 2*j**

a*b = 14 and c*b = 16

Where a= 3i+j and c = 5i-2j.

Let b = xi+yj.

Therefore a*b = (3i+j) *(xi+yj) = 3xi^2+yj^2 = 3x+y = 14.

c*b = (5i-2j)*(xi+yj) = 5xi^2-2yj^2= 5x-2y = 16

Therefore we have to solve the equations for x and y:

3x+y = 14....(1)

5x-2y = 16...(2)

2eq(1) +eq(2) gives: 6x+5y = 2*14+16 = 44.

11x = 44

So x= 44/11= 4.

Put x= 4 in 3x+y = 14.

So 3*4 +y = 14.. So y = 14-12 = 2.

So x= 2 and y = 2.