Question

Exercise 1. Find the area of the polygon in R3 with corners (-7, 7), (5,5), (-5, -6) and (3, -4) Exercise 2. Find the 3-volumne of the 3-parallelepiped defined by the vectors Exercise 3. Consider those 4 x 4 matriceis whose entries are all 1, -1 or 0. What is the maximal value of the determinant of a matrix of this type? Justify your answer algebraically and geometrically. Give an example of a matrix whose determinant has this maximal value.

ANSWER ALL QUESTIONS/EXERCISES IN DETAIL. DO NOT ANSWER JUST ONE. PLEASE, ANSWER ALL.

Answer 1

Exercise To find the area of polygon with Corners at (-7,7), (5,5) (-5,-6) (3,-4) (-7,720 (1515) 2 3 8 B - (3-4) (-51-6) Note that all corners are in e same 2 plane. so it is enough to find the area in PR² We into two parts devide thea area Ý A ABC () ADC Area of SABC } x base x perpendicular distance We take base AB so perpendicular distance AB from is a distance of C Eqn of ĀB :- -5-6 3,- y+6 275 -4+6 3+5 8 y + 48 = 22 +0 - 38 -0 A 2x-84 Cy 서 a-44 - 19- length of AB - 5(-6-(-4)) {+ (-5-3) -368 - 4+64 and ť distance from @ (5.5) to line AB Тg 5-4.5 - 19 ritib 5-20 -19 Viz H 34 17 : Area of a ABC - 1/2 x 168 x 3 in Ź Jux こ *34 - 2/2 x 34 Sq. wmi't. = 34 Similarly we find out 4 АСР area of As (-5,-6) es (55) D= (-7,7) line egn of CD y-7 x-7 5-7 5+7 12 y D 84 = -28 +14 2x +12y - 70 O - xt by - 35 = Length of a (5+7)2+(5-7)2 - 148 144 +4 CD Į distance from A to -5+61-6) - 35 1+36 45 76 537 38 76 1. z 148 :. Area of SACD = 2) 148 (74 38x54x37 37 76 sq.unip Hence area of poly gon | ABCD is (76 + 34) sq. conit 110 squit