the triangle with the vertices \( P=(2,2,7), Q=(4,5,4) \), and \( R=(2,7,10) \) in

To find the area of the triangle defined by the points \( P(2,2,7) \), \( Q(4,5,4) \), and \( R(2,7,10) \), one can use the cross product of the vectors formed by the sides of the triangle. First, create vectors \( \overrightarrow{PQ} \) and \( \overrightarrow{PR} \) from point \( P \) to points \( Q \) and \( R \). Then, the area can be calculated as half the magnitude of the cross product of these two vectors—voilà, geometry at work! For those curious about geometric shapes in three dimensions, the principles behind calculating the area of a triangle extend far beyond this scenario—consider exploring tetrahedrons or even more complex polyhedra. Shapes like these involve fascinating concepts of volume and surface area, making for a delightful dive into the realm of spatial geometry.