# vectorsFind the vector v if u*v=15, w*v=17, u=5i+2j, w=i-j.

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### 2 Answers

We have to find vector v if u*v=15, w*v=17 and u=5i+2j, w=i-j

Let v = ai + bj

u*v = 15 = 5a + 2b ...(1)

w*v = 17 = a - b ...(2)

(1) + 2*(2)

=> 5a + 2b + 2a - 2b = 34 + 15

=> 7a = 49

=> a = 7

17 = 7 - b

=> b = -10

**The vector v = 7i - 10j**

A bi-dimensional vector is written as:

v = x*i + y*j

We'll use the dot product definition.

u*v = (5i+2j)(xi + yj)

u*v = 5xi^2 + 5yij + 2xij + 2yj^2

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

u*v = 5x + 2y

Substituting, we'll get the following system:

5x + 2y = 15 (1)

w*v = (i-j)(xi + yj)

w*v = x - y

x - y = 17 => x = 17 + y (2)

We'll substitute (2) in (1):

5(17 + y) + 2y = 15

We'll remove the brackets:

85 + 5y + 2y = 15

7y = 15 - 85

7y = -70

y = -10

x = 17 - 10

x = 7

So, the requested vector is:** **

**v=7*i - 10*j**