Objective: Find the value of $34(a + b + c)$, where $P(a, b, c)$ is the foot of the perpendicular from the origin to the line L.

Find Direction Vector of Line L

The line L is perpendicular to the vectors $\vec{v_1} = 2\hat{i} + 2\hat{j} + \hat{k}$ and $\vec{v_2} = \hat{i} + 2\hat{j} + 2\hat{k}$. The direction vector $\vec{d}$ of line L is the cross product of $\vec{v_1}$ and $\vec{v_2}$.

$ \vec{d} = \vec{v_1} \times \vec{v_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 2 & 1 \\ 1 & 2 & 2 \end{vmatrix} $

$ \vec{d} = \hat{i}(2 \cdot 2 - 1 \cdot 2) - \hat{j}(2 \cdot 2 - 1 \cdot 1) + \hat{k}(2 \cdot 2 - 2 \cdot 1) $

$ \vec{d} = \hat{i}(4 - 2) - \hat{j}(4 - 1) + \hat{k}(4 - 2) = 2\hat{i} - 3\hat{j} + 2\hat{k} $

The direction vector is $\vec{d} = (2, -3, 2)$.

Determine Equation of Line L

Line L passes through the point $(1, 1, 1)$ and has the direction vector $(2, -3, 2)$. The parametric equations for line L are:

$ x = 1 + 2t $ $ y = 1 - 3t $ $ z = 1 + 2t $

Any point on the line L can be represented as $(1 + 2t, 1 - 3t, 1 + 2t)$.

Identify Coordinates of Foot of Perpendicular P

The point $P(a, b, c)$ is the foot of the perpendicular from the origin $(0, 0, 0)$ to the line L. Therefore, the vector $\vec{OP} = a\hat{i} + b\hat{j} + c\hat{k}$ is perpendicular to the direction vector $\vec{d}$ of the line.

Coordinates of P are $(a, b, c) = (1 + 2t, 1 - 3t, 1 + 2t)$.

The vector $\vec{OP} = (1 + 2t)\hat{i} + (1 - 3t)\hat{j} + (1 + 2t)\hat{k}$.

The condition for perpendicularity is $\vec{OP} \cdot \vec{d} = 0$:

$ ((1 + 2t)\hat{i} + (1 - 3t)\hat{j} + (1 + 2t)\hat{k}) \cdot (2\hat{i} - 3\hat{j} + 2\hat{k}) = 0 $

$ 2(1 + 2t) - 3(1 - 3t) + 2(1 + 2t) = 0 $

$ 2 + 4t - 3 + 9t + 2 + 4t = 0 $

$ 17t + 1 = 0 \implies t = -\frac{1}{17} $

Now, substitute $t = -1/17$ to find the coordinates of P:

$ a = 1 + 2(-\frac{1}{17}) = 1 - \frac{2}{17} = \frac{15}{17} $ $ b = 1 - 3(-\frac{1}{17}) = 1 + \frac{3}{17} = \frac{20}{17} $ $ c = 1 + 2(-\frac{1}{17}) = 1 - \frac{2}{17} = \frac{15}{17} $

So, $P(a, b, c) = (\frac{15}{17}, \frac{20}{17}, \frac{15}{17})$.

Calculate Final Value

We need to find the value of $34(a + b + c)$.

First, calculate the sum $a + b + c$:

$ a + b + c = \frac{15}{17} + \frac{20}{17} + \frac{15}{17} = \frac{15 + 20 + 15}{17} = \frac{50}{17} $

Now, multiply the sum by 34:

$ 34(a + b + c) = 34 \times \frac{50}{17} $

$ 34(a + b + c) = \frac{34}{17} \times 50 = 2 \times 50 = 100 $

The value of $34(a + b + c)$ is 100.