Given that,

We need to find vector .

Let , where x, y, z be any scalars.

Now, for :

Comparing Left Hand Side and Right Hand Side, we get

From coefficient of ⇒ z – y = 0 …(i)

From coefficient of ⇒ -(z – x) = 1

⇒ x – z = 1 …(ii)

From coefficient of ⇒ y – x = -1

⇒ x – y = 1 …(iii)

Also, for :

Since, , as and other dot multiplication is zero. We get,

x + y + z = 3 …(iv)

Now, add equations (ii) and (iii), we get

(x – z) + (x – y) = 1 + 1

⇒ x + x – y – z = 2

⇒ 2x – y – z = 2 …(v)

Add equations (iv) and (v), we get

(x + y + z) + (2x – y – z) = 3 + 2

⇒ x + 2x + y – y + z – z = 5

⇒ 3x = 5

Put value of x in equation (iii), we get

Equation (iii) ⇒ x – y = 1

Put this value of y in equation (i), we get

Equation (i) ⇒ z – y = 0

Since,

By putting the values of x, y and z, we get

Thus, we have found the vector .