Let the vertices Q and R of the triangle PQR lie on the line 5x+3=2y−1=3z+4, QR = 5, and the coordinates of the point P be (0,2,3). If the area of the triangle PQR is nm, then:
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Step-by-Step Solution
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 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 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 AP and 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 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 ∣d∣∣AP×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.