Let the vertices Q and R of the triangle PQR lie on the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$, QR = 5, and the coordinates of the point P be $(0{,}2{,}3)$. If the area of the triangle PQR is $\tfrac{m}{n}$, then:

Step 1: Identify the line and point

The problem provides the equation of the line $L$ on which vertices Q and R lie, and the coordinates of point P. We need to find the area of triangle PQR.

Step 2: Find a point on the line and its direction vector

From the symmetric form of the line equation, we can identify a point $A$ on the line and its direction vector $\vec{d}$. The point $A$ is obtained by setting the numerators to zero, and the direction vector components are the denominators.

Step 3: Calculate vector AP

To find the perpendicular distance from point P to the line L, we first calculate the vector $\vec{AP}$ connecting point A on the line to point P.

Step 4: Calculate the cross product and its magnitude

The magnitude of the cross product of $\vec{AP}$ and $\vec{d}$ is used to find the area of the parallelogram formed by these two vectors. This value is crucial for calculating the perpendicular distance.

Step 5: Calculate the magnitude of the direction vector

We need the magnitude of the direction vector $\vec{d}$ to calculate the perpendicular distance from point P to the line L.

Step 6: Calculate the perpendicular distance (height)

The perpendicular distance $h$ from point P to the line L (which is the height of the triangle PQR with respect to base QR) is given by the formula $\frac{|\vec{AP} \times \vec{d}|}{|\vec{d}|}$.

Step 7: Calculate the area of triangle PQR

The area of triangle PQR is calculated using the standard formula: half times base times height. Here, the base is QR, given as 5, and the height is the perpendicular distance $h$ we just calculated.

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