To solve this problem, we can use the concept of the equation of a line and the distance formula.

Let's start by finding the equation of the line L. Since the line is parallel to the vector i j k, we can write the equation of the line in vector form as:

r = (1, 2, 4) + t(i, j, k)

Here, (1, 2, 4) represents the given point on the line, and (i, j, k) represents the direction vector of the line. t is a parameter that varies over the real numbers.

Now, let's find the point where the line L meets the xy-plane. The xy-plane can be represented by the equation z = 0. To find the point of intersection, we substitute z = 0 into the equation of the line and solve for t:

0 = 4 + tk

Solving for t, we get:

t = -4/k

Now, substitute this value of t back into the equation of the line to find the coordinates of the point of intersection:

r = (1, 2, 4) + (-4/k)(i, j, k)
= (1 - 4/k, 2 - 4/k, 4 - 4/k)
= (1 - 4/k, 2 - 4/k, 4 - 4/k)

The coordinates of the point of intersection are (1 - 4/k, 2 - 4/k, 4 - 4/k).

To find the distance between the origin and point P, we can use the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

Let's plug in the coordinates of P (1 - 4/k, 2 - 4/k, 4 - 4/k) and the origin (0, 0, 0) into the distance formula:

d = sqrt((1 - 4/k - 0)^2 + (2 - 4/k - 0)^2 + (4 - 4/k - 0)^2)
= sqrt((1 - 4/k)^2 + (2 - 4/k)^2 + (4 - 4/k)^2)
= sqrt((1 - 4/k)^2 + (2 - 4/k)^2 + (4 - 4/k)^2)

Now, we need to simplify this expression to find the distance between the origin and point P. To do this, we expand the squares and simplify:

d = sqrt((1 - 4/k)^2 + (2 - 4/k)^2 + (4 - 4/k)^2)
= sqrt((1 - 8/k + 16/k^2) + (4 - 16/k + 16/k^2) + (16 - 16/k + 16/k^2))
= sqrt(1 - 8/k + 16/k^2 + 4 - 16/k + 16/k^2 + 16 - 16/k + 16/k^2)
= sqrt(37 - 40/k + 48/k^2)

To find the value of k that minimizes the distance, we can take the derivative of the expression with respect to k and set