Given Information:
The dot product of a vector with the vectors (i^ j^-3k^), (i^ 3j^-2k^), and (2i^ j^ 4k^) are 0, 5, and 8 respectively.

To Find:
The vector.

Solution:

Step 1: Introduction
We are given three vectors and their dot products with an unknown vector. We need to find the unknown vector.

Step 2: Understanding Dot Product
The dot product of two vectors is defined as the product of their magnitudes and the cosine of the angle between them. Mathematically, the dot product of two vectors A and B is given by:

A · B = |A| |B| cosθ

where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them.

Step 3: Dot Product with the Given Vectors
Let the unknown vector be denoted as (xi^ + yj^ + zk^). We can now write the dot product equations using the given information:

(i^ j^-3k^) · (xi^ + yj^ + zk^) = 0
(i^ 3j^-2k^) · (xi^ + yj^ + zk^) = 5
(2i^ j^ 4k^) · (xi^ + yj^ + zk^) = 8

Step 4: Expanding the Dot Product Equations
Expanding the dot product equations using the properties of dot product, we get:

x - 3y = 0
x + 3y - 2z = 5
2x + y + 4z = 8

Step 5: Solving the System of Equations
We have a system of three linear equations with three variables (x, y, z). We can solve this system to find the values of x, y, and z.

Solving the system of equations, we get:
x = 3
y = 1
z = 2

Step 6: Finding the Unknown Vector
Now that we have the values of x, y, and z, we can substitute them into the equation of the unknown vector:

(xi^ + yj^ + zk^) = (3i^ + j^ + 2k^)

Thus, the unknown vector is (3i^ + j^ + 2k^).

Conclusion:
The unknown vector is (3i^ + j^ + 2k^).