scalar triple product - 2025 Penn Bowl

Question

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]

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Summary


2025 Penn Bowl - BU	11/01/25	Y	4	10.00	75%	25%	0%
2025 Penn Bowl - CWRU	11/01/25	Y	3	10.00	100%	0%	0%
2025 Penn Bowl - Mainsite (UPenn)	11/01/25	Y	6	10.00	83%	17%	0%
2025 Penn Bowl - Murray State	11/08/25	Y	2	5.00	50%	0%	0%
2025 Penn Bowl - Northwestern	10/25/25	Y	4	15.00	50%	75%	25%
2025 Penn Bowl - Online	11/08/25	Y	2	5.00	50%	0%	0%
2025 Penn Bowl - Ottawa	11/01/25	Y	7	7.14	57%	14%	0%
2025 Penn Bowl - Stanford	10/25/25	Y	4	7.50	50%	25%	0%
2025 Penn Bowl - UK	11/01/25	Y	9	11.11	67%	44%	0%
2025 Penn Bowl - UNC	11/01/25	Y	3	3.33	33%	0%	0%
2025 Penn Bowl - WashU	11/01/25	Y	2	15.00	100%	50%	0%

Data


Brown	Brandeis	10	0	10	20
Yale	I've been running into you in my lead plumbocene what could have benzene	0	0	0	0
Harvard	MIT A	0	0	10	10
Ohio State A	Michigan A	0	0	10	10
CWRU	Ohio State B	0	0	10	10
Pittsburgh	Michigan C	0	0	10	10
MIT B	RIT	0	0	10	10
Columbia B	JHU B	0	0	10	10
Mount Sinai	NYU A	10	0	0	10
NYU B	John Jay	0	0	10	10
Columbia A	Swarthmore	0	0	10	10
Oklahoma	Central Oklahoma	0	0	0	0
Texas A	Texas B	0	0	10	10
Purdue B	Illinois	10	0	0	10
Indiana	UChicago A	10	10	10	30
Northwestern B	Northwestern A	0	0	10	10
Purdue A	UChicago C	10	0	0	10
rusty shackleford	New Young Whigs of America	0	0	0	0
abg academy	Wake Up Paddington A Washington Mystery	0	0	10	10
Aweille Kevin Fan osti continue comme ça	#OpenToBuzz	0	0	0	0
Guelph perfomative male competition	Carleton A	0	0	0	0
McGill B	Toronto E	0	0	10	10
McGill Fartlets	Ottawa 67s	0	0	10	10
Toronto Murder on the Rider Express	Toronto Sunshine Bubbles	0	0	10	10
Toronto G	Toronto Operation PBSuccess	10	0	10	20
Queen's is monitoring the situation	Waterloo you are crooked	0	0	0	0
Berkeley A	Berkeley B	10	0	10	20
Free Win	In Search of Lost Clues	0	0	0	0
Stanford A	Berkeley C	0	0	10	10
The best time to buzz was twenty clues ago. The second best time to buzz is now	Stanford B	0	0	0	0
Alas poor Warwick, I knew him Edinburgh	1. Wawa 2. Blank Street 151. Black Sheep at Kings Cross rail	0	0	0	0
Hail to the Sheaf	Burnaburiash's Lapis Lazuli	10	0	10	20
Knorr Stock Clue	Of Mice and Penn	0	0	10	10
Bristold	Paul Quifford, or, It Was a Dark and Stormy Tournament	0	0	0	0
Penn and the Art of Tossup-Bonus cycle Maintenance	question mark	10	0	10	20
Quiztian Bale in Ratman Begins	Galactic Ombudsmen	0	0	10	10
The Bit and the Penndulum	Chimp Brisket	10	0	10	20
Washed Scrubs	Coughing Baby	0	0	0	0
To buzz is to be perceived	why is it called a score clue if nobody scores points on it	10	0	10	20
Maryland A	JHU A	0	0	10	10
Maryland B	Maryland C	0	0	10	10
UNC B	Planck Sinatra	0	0	0	0
VT A	Wise Duke	0	0	0	0
VT B	UNC A	0	0	10	10
WashU Bouba	Truman State	0	0	10	10
WashU Kiki	VJ Edgecombe School of Interior Design - Edwardsville	10	0	10	20