To find the distance between the point $Q(3, 4, 1)$ and the given line described by the parametric equations:

$x = -4t + 4, \quad y = 3 - t, \quad z = t - 3$

we can use the formula for the distance from a point to a line in 3D space:

$\text{Distance} = \frac{\| \mathbf{PQ} \times \mathbf{d} \|}{\|\mathbf{d}\|}$

where:

• $\mathbf{P}$ is a point on the line (obtained by plugging any value of $t$ into the parametric equations; let's use $t = 0$ for simplicity).
• $\mathbf{Q} = (3, 4, 1)$ is the given point.
• $\mathbf{d}$ is the direction vector of the line (coefficients of $t$ in the parametric equations).

Step 1: Identify a point on the line ($\mathbf{P}$) and the direction vector ($\mathbf{d}$).

For $t = 0$: $\mathbf{P} = (x_0, y_0, z_0) = (4, 3, -3)$

The direction vector $\mathbf{d}$ is: $\mathbf{d} = (-4, -1, 1)$

Step 2: Find vector $\mathbf{PQ}$.

$\mathbf{PQ} = \mathbf{Q} - \mathbf{P} = (3 - 4, 4 - 3, 1 - (-3)) = (-1, 1, 4)$

Step 3: Calculate the cross product $\mathbf{PQ} \times \mathbf{d}$.

$\mathbf{PQ} \times \mathbf{d} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -1 & 1 & 4 \\ -4 & -1 & 1 \end{vmatrix}$

$= \mathbf{i}(1 \times 1 - 4 \times -1) - \mathbf{j}(-1 \times 1 - 4 \times -4) + \mathbf{k}(-1 \times -1 - 1 \times -4)$

$= \mathbf{i}(1 + 4) - \mathbf{j}(-1 + 16) + \mathbf{k}(1 + 4)$

$= \mathbf{i}(5) - \mathbf{j}(15) + \mathbf{k}(5)$

$\mathbf{PQ} \times \mathbf{d} = (5, -15, 5)$

Step 4: Find the magnitude of $\mathbf{PQ} \times \mathbf{d}$.

$\| \mathbf{PQ} \times \mathbf{d} \| = \sqrt{5^2 + (-15)^2 + 5^2} = \sqrt{25 + 225 + 25} = \sqrt{275} = 5\sqrt{11}$

Step 5: Find the magnitude of $\mathbf{d}$.

$\|\mathbf{d}\| = \sqrt{(-4)^2 + (-1)^2 + 1^2} = \sqrt{16 + 1 + 1} = \sqrt{18} = 3\sqrt{2}$

Step 6: Calculate the distance.

$\text{Distance} = \frac{5\sqrt{11}}{3\sqrt{2}} = \frac{5\sqrt{22}}{6}$

So, the exact distance is $\frac{5\sqrt{22}}{6}$.

Would you like more details or have any questions about this process? Here are some related questions:

1. How can you use the dot product to check if two vectors are perpendicular?
2. What is the significance of the direction vector in the parametric equation of a line?
3. How do you derive the cross product in a 3D vector space?
4. Can you describe how to convert parametric equations of a line to a symmetric form?
5. What other methods can be used to find the distance between a point and a line in space?

Tip: Always verify the vector magnitudes and cross product carefully, as small mistakes can lead to incorrect results in distance calculations.