The perpendicular distance, of the line $\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z+3}{2}$ from the point $P(2,-10{,}1)$, is:

Step 1: Identify line equation and point

First, we identify the given line in symmetric form. From the equation, we can extract a point $P_0$ on the line and the direction vector $\vec{d}$ of the line. We are also given the point $P$ from which we need to find the perpendicular distance.

Step 2: Find vector from point on line to given point

Next, we calculate the vector $\vec{P_0P}$ by subtracting the coordinates of the point $P_0$ on the line from the coordinates of the given point $P$. This vector connects a point on the line to the external point.

Step 3: Calculate cross product magnitude

To find the perpendicular distance, we use the formula involving the cross product. We compute the cross product of the vector $\vec{P_0P}$ and the direction vector $\vec{d}$. Then, we find the magnitude of this resulting cross product vector.

Step 4: Calculate magnitude of direction vector

We also need the magnitude of the direction vector $\vec{d}$ for the distance formula. We calculate this by taking the square root of the sum of the squares of its components.

Step 5: Calculate perpendicular distance

Finally, we apply the formula for the perpendicular distance from a point to a line: $D = \frac{|\vec{P_0P} \times \vec{d}|}{|\vec{d}|}$. We substitute the calculated magnitudes to find the final distance.

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