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Use a table saw to rip the full plywood sheets to make the bottom and sides of the vanity. Measure and cut carefully. Sloppy cuts or inaccurate measurements will throw off the entire structure.
With the plywood cut to size lengthwise, use a circular saw to finish cutting out the individual panel pieces. Use a guide, such as a metal carpenter's square clamped to the wood, to ensure a straight cut.
The contemporary design for this vanity uses dado joints for the plywood instead of butt joints for a cleaner look. Dado joints also add strength to the overall piece. Line up the side panels side by side and clamp them securely together with long bar clamps. Also clamp them to the work surface below.
Measure and mark the location of the dado cuts in the side panels, then clamp a guide in place to keep the router on track. Run the router along the guide to make a channel about 3/8" deep.
Sand down the rough edges of the dado channels.
To assemble the plywood panels, run a bead of wood glue down the dado channel on one of the side pieces and insert the bottom panel. Tap the pieces together with a mallet, then reinforce the joint with finish nails. Repeat for the other side panel.
Measure and cut three plywood nailers to fit snugly between the side panels. Two will go at the top and bottom of the back of the vanity to attach it to the wall; the other will go at the top front of the vanity to support the countertop and sinks.
Use a drill and a pocket hole jig to drill pocket holes in the nailers. Pocket holes start in the side of a piece of wood and come out the cut end, making it possible to "toenail" the piece with hidden screws. Drill two pocket holes at each end of each nailer.
Attach the nailers to the side panels with pocket hole screws.
Use a miter saw to cut horizontal plywood rails to fit the top and bottom of the vanity front, plus three vertical stiles. Attach the rails and the two side stiles with wood glue and finish nails, covering the cut edges of wood.
Drill pocket holes at both ends of the center stile. Clamp it in place and attach with pocket hole screws, keeping the screws on the inside of the cabinet.
Measure and cut plywood for the doors. Rip the wood stock on a table saw and use a circular saw to cut them to the correct height. Sand down the rough edges.
The vanity design calls for concealed hinges in the doors. To install the hinges, use a drill with a Forstner bit and a self-centering drill guide to create wide, round recesses in the doors.
Screw the hinges in place on the doors and the inside of the cabinet. Finish the piece as desired.
To install the vanity, remove the doors and set them aside. Drill pilot holes and drive screws through the nailers to attach the cabinet to the wall studs.