1. Key Concepts and Formulas

• Equation of a Line in 3D (Cartesian Form): A line passing through a point $(x_1, y_1, z_1)$ with a direction vector $(a, b, c)$ can be written as: $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} = k$ where $k$ is a scalar parameter.
• Perpendicularity of Vectors: If two vectors $\vec{v_1} = (a_1, b_1, c_1)$ and $\vec{v_2} = (a_2, b_2, c_2)$ are perpendicular, their dot product is zero: $\vec{v_1} \cdot \vec{v_2} = a_1a_2 + b_1b_2 + c_1c_2 = 0$
• Distance Formula in 3D: The distance between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ is given by: $P_1P_2 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

2. Step-by-Step Solution

Step 1: Determine the Equation of the Line

We are given that the line passes through points $A(2, -5, 11)$ and $B(-6, 7, -5)$. To find the equation of the line, we first need its direction vector. We can use the vector $\vec{BA}$: $\vec{d_L} = \vec{BA} = (2 - (-6), -5 - 7, 11 - (-5)) = (8, -12, 16)$ We can use point $A(2, -5, 11)$ and the direction vector $\vec{d_L} = (8, -12, 16)$ to write the Cartesian equation of the line: $\frac{x - 2}{8} = \frac{y - (-5)}{-12} = \frac{z - 11}{16}$ $\frac{x - 2}{8} = \frac{y + 5}{-12} = \frac{z - 11}{16}$ Reasoning: The direction vector defines the orientation of the line in space, and a point on the line gives a starting reference. These two components are sufficient to define the line's equation.

Step 2: Express the Coordinates of Point Q

Since $Q$ is a point on the line, we can represent its coordinates using a scalar parameter, say $k$. From the line equation, we set each fraction equal to $k$: $\frac{x - 2}{8} = k \implies x = 8k + 2$ $\frac{y + 5}{-12} = k \implies y = -12k - 5$ $\frac{z - 11}{16} = k \implies z = 16k + 11$ So, the general coordinates of point $Q$ on the line are $(8k + 2, -12k - 5, 16k + 11)$. Reasoning: This parametric form allows us to represent any point on the line and later find the specific point $Q$ that satisfies the perpendicularity condition by determining the value of $k$.

Step 3: Formulate the Vector $\vec{RQ}$

We are given the point $R(1, 7, 6)$ and the general coordinates of $Q(8k + 2, -12k - 5, 16k + 11)$. The vector $\vec{RQ}$ is found by subtracting the coordinates of $R$ from $Q$: $\vec{RQ} = ( (8k + 2) - 1, (-12k - 5) - 7, (16k + 11) - 6 )$ $\vec{RQ} = (8k + 1, -12k - 12, 16k + 5)$ Reasoning: The vector $\vec{RQ}$ connects point $R$ to the point $Q$ on the line. For $Q$ to be the foot of the perpendicular from $R$, this vector must be perpendicular to the line itself.

Step 4: Apply the Perpendicularity Condition to Find $k$

Since $Q$ is the foot of the perpendicular from $R$ to the line, the vector $\vec{RQ}$ must be perpendicular to the direction vector of the line, $\vec{d_L} = (8, -12, 16)$. The dot product of two perpendicular vectors is zero: $\vec{RQ} \cdot \vec{d_L} = 0$ $(8k + 1)(8) + (-12k - 12)(-12) + (16k + 5)(16) = 0$ Expand and solve for $k$: $(64k + 8) + (144k + 144) + (256k + 80) = 0$ Combine the $k$ terms: $(64 + 144 + 256)k = 464k$ Combine the constant terms: $8 + 144 + 80 = 232$ The equation becomes: $464k + 232 = 0$ $464k = -232$ $k = -\frac{232}{464} = -\frac{1}{2}$ Reasoning: This step uses the geometric property of perpendicularity to find the unique value of $k$ that corresponds to the foot of the perpendicular.

Step 5: Find the Coordinates of Point Q

Substitute the value of $k = -1/2$ back into the expressions for the coordinates of $Q$: $Q_x = 8\left(-\frac{1}{2}\right) + 2 = -4 + 2 = -2$ $Q_y = -12\left(-\frac{1}{2}\right) - 5 = 6 - 5 = 1$ $Q_z = 16\left(-\frac{1}{2}\right) + 11 = -8 + 11 = 3$ So, the coordinates of the foot of the perpendicular, $Q$, are $(-2, 1, 3)$. Reasoning: With the value of $k$ determined, we can now find the exact coordinates of the specific point $Q$ on the line.

Step 6: Calculate the Length of the Line Segment PQ

We need to find the distance between point $P(10, -2, -1)$ and point $Q(-2, 1, 3)$. Using the 3D distance formula: $PQ = \sqrt{(x_Q - x_P)^2 + (y_Q - y_P)^2 + (z_Q - z_P)^2}$ $PQ = \sqrt{(-2 - 10)^2 + (1 - (-2))^2 + (3 - (-1))^2}$ $PQ = \sqrt{(-12)^2 + (3)^2 + (4)^2}$ $PQ = \sqrt{144 + 9 + 16}$ $PQ = \sqrt{169}$ $PQ = 10$ Reasoning: The final step is a direct application of the distance formula to find the required length between the two specified points.

3. Common Mistakes & Tips

• Direction Vector Calculation: Ensure the direction vector is calculated correctly by subtracting coordinates in a consistent order (e.g., $B-A$ or $A-B$). A scalar multiple of the direction vector is also valid.
• Sign Errors: Be careful with signs, especially when dealing with negative coordinates and subtraction in the dot product and distance formulas.
• Parameter $k$ Substitution: Double-check the substitution of the calculated $k$ value back into the parametric equations of the line to find the correct coordinates of $Q$.

4. Summary

The problem required finding the length of a line segment $PQ$, where $Q$ is the foot of the perpendicular from a given point $R$ to a line defined by two other points. We first established the parametric equation of the line. Then, we expressed the coordinates of $Q$ in terms of a parameter $k$. By using the condition that the vector $\vec{RQ}$ is perpendicular to the line's direction vector, we found the value of $k$ and thus the coordinates of $Q$. Finally, we used the 3D distance formula to calculate the length of $PQ$.

5. Final Answer

The final answer is $\boxed{10}$.