Given Data:

Vectors:

Their Dot products are 0, 5 and 8.

Calculation:

Let the required vector be,

Now,

x + y -3z =0 …Eq. 1

Similarly

x + 3y - 2z = 5 …Eq. 2

2x + y + 4z = 8 …Eq. 3

Subtract Eq. 1 from Eq. 2

(x + 3y - 2z) – (x + y -3z) = 5 - 0

⇒ 2y + z = 5 …Eq. 4

Subtract Eq. 3 from (2 × Eq. 2)

2(x + 3y - 2z) - 2x + y + 4z = (2 × 5) - 8

5y – 8z = 2 …Eq. 5

Adding Eq. 5 with (8 × Eq. 4)

8(2y + z) + (5y – 8z) = 8 × 5 + 2

⇒ 21y = 42

⇒ y = 2

From Eq. 5,

5×2 - 8z = 2

⇒ z = 1

From Eq. 1

x + y -3z =0

⇒ x + 2 – 3×1 = 0

⇒ x = 1