Reduce to canonical form the following quadratic forms on r 3 :
a) q(x) = x12 + x 22 + 3x32+ 4x1x2 + 2x1x3 + 2x2x3;
b) q(x) = 2x1x2 − 6x2x3 + 2x1x3.
the form q = a1y12+ a2y22 + · · · + anyn2 , where y1, y2, . . , yn are new unknowns, are called canonical

Aflask with vinegar in it has a mass of 160 grams. a balloon with baking soda in it has a mass of 40 grams. the balloon is attached to the flask to seal the opening and the vinegar and baking soda mixes. the balloon inflates to a large volume. what will the total mass of the balloon and flask be after the balloon inflates? explain. a) less than 200 grams because the solid baking soda disappears. b) 200 grams, because all the atoms remain in the balloon or flask. c) more than 200 grams because the size of the balloon is so much larger. d) less than 200 grams because gases such as the one in the balloon are lighter than solids and liquids.