Step 1: Finding the Foot of the Perpendicular

The line is given by the vector equation $\vec{r} = (-\hat{i} + 3\hat{j} + \hat{k}) + \lambda(2\hat{i} + 3\hat{j} - \hat{k})$.

A general point on the line has coordinates $(x, y, z)$, where:

• $x = -1 + 2\lambda$
• $y = 3 + 3\lambda$
• $z = 1 - \lambda$

Let the foot of the perpendicular from the point $P(5, 4, 2)$ be $F(\alpha, \beta, \gamma)$. This means $F$ is a point on the line, so $\alpha = -1 + 2\lambda$, $\beta = 3 + 3\lambda$, and $\gamma = 1 - \lambda$ for a specific value of $\lambda$.

The vector $\vec{PF}$ is calculated as $F - P$: $ \vec{PF} = (\alpha - 5)\hat{i} + (\beta - 4)\hat{j} + (\gamma - 2)\hat{k} $ Substituting the expressions for $\alpha, \beta, \gamma$: $ \vec{PF} = (-1 + 2\lambda - 5)\hat{i} + (3 + 3\lambda - 4)\hat{j} + (1 - \lambda - 2)\hat{k} $ $ \vec{PF} = (2\lambda - 6)\hat{i} + (3\lambda - 1)\hat{j} + (-\lambda - 1)\hat{k} $

The direction vector of the line is $\vec{d} = 2\hat{i} + 3\hat{j} - \hat{k}$.

Since $\vec{PF}$ is perpendicular to the line's direction vector $\vec{d}$, their dot product must be zero: $\vec{PF} \cdot \vec{d} = 0$.

$ (2\lambda - 6)(2) + (3\lambda - 1)(3) + (-\lambda - 1)(-1) = 0 $ Expanding this equation: $ (4\lambda - 12) + (9\lambda - 3) + (\lambda + 1) = 0 $ Combining terms: $ 14\lambda - 14 = 0 $ Solving for $\lambda$: $ \lambda = 1 $

Now, we find the coordinates $(\alpha, \beta, \gamma)$ by substituting $\lambda = 1$ back into the general point equations:

• $ \alpha = -1 + 2(1) = 1 $
• $ \beta = 3 + 3(1) = 6 $
• $ \gamma = 1 - 1 = 0 $

Thus, the coordinates of the foot of the perpendicular are $(\alpha, \beta, \gamma) = (1, 6, 0)$.

Step 2: Calculating the Projection Length

We need to find the length of the projection of the vector $\vec{v} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$ onto the vector $\vec{w} = 6\hat{i} + 2\hat{j} + 3\hat{k}$.

Using the coordinates found, the vector $\vec{v}$ is $1\hat{i} + 6\hat{j} + 0\hat{k}$.

The formula for the length of the projection of vector $\vec{v}$ onto vector $\vec{w}$ is:

$ \text{Projection Length} = \frac{|\vec{v} \cdot \vec{w}|}{||\vec{w}||} $

Calculate the dot product $\vec{v} \cdot \vec{w}$:

$ \vec{v} \cdot \vec{w} = (1)(6) + (6)(2) + (0)(3) = 6 + 12 + 0 = 18 $

Calculate the magnitude of vector $\vec{w}$:

$ ||\vec{w}|| = \sqrt{6^2 + 2^2 + 3^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 $

Now, compute the projection length:

$ \text{Projection Length} = \frac{|18|}{7} = \frac{18}{7} $