In three dimensions, the Gram determinant is equivalent to the square of this mathematical concept, according to Lagrange’s identity. For 10 points each:
[10m] Name this mathematical concept, whose “scalar” form uses vectors A, B, and C to define the volume of a parallelepiped.
ANSWER: scalar triple product [or box product; or mixed product; or triple scalar product; prompt on product; reject “vector triple product” or “dot product” or “cross product”]
[10h] In a formula sometimes named for Lagrange, the vector triple product of A, B, and C may be expanded as this expression of their dot and scalar products. Give this expression, or the common mnemonic used to summarize it.
ANSWER: ACB minus ABC [or BAC minus CAB; or A dot C times B minus A dot B times C; or B times A dot C minus C times A dot B; accept answers that insert parentheses into any of those; accept dot product in place of “dot”; reject answers that mention “cross” or “cross product”]
[10e] Lagrange also developed a method of using namesake “multipliers” to find these values of functions. Critical points are identified as these values by the second derivative test.
ANSWER: extrema [or extremums or extreme values; accept maxima or minima or maximums or minimums or maxima and minima or maximums and minimums]
<Benjamin McAvoy-Bickford, Other Science>