A point $$P\left( {1,\,\,\,3,\,\,\, - 5} \right)$$ is translated by $$2\widehat i + 3\widehat j - 4\widehat k$$ and then rotated counter clockwise by $${90^ \circ }$$ about the $$z$$-axis. The new position of the point is

The figure below represents a triangle $$PQR$$ with initial coordinates of the vertices as $$P(1,3),$$ $$Q(4,5)$$ and $$R(5,3.5).$$ The triangle is rotated in the $$X-Y$$ plane about the vertex $$P$$ by angle in clockwise direction. If sin $$\theta = 0.6$$ and cos $$\theta = 0.8$$, the new coordinates of the vertex $$Q$$ are

A triangular facet in a $$CAD$$ model has vertices: $${P_1}\left( {0,0,0} \right);\,\,{P_2}\left( {1,1,0} \right)$$ and $$\,{P_3}\left( {1,1,1} \right).$$ The area of the facet is

In a $$CNC$$ milling operation, the tool has to machine the circular arc from point $$(20, 20)$$ to $$(10, 10)$$ at sequence number $$5$$ of the $$CNC$$ part program. If the center of the arc is at $$(20, 10)$$ and the machine has incremental mode of defining position coordinates, the correct tool path command is