MAC 2313 HANDOUT 1Note: These exercises are usually more involved than those from reg-ular homework. They are intended for those who are interested ingetting a more in-depth practice with vectors.1.) Let AA1, BB1, and CC1be the medians of a triangle ABC. Provethat−→AB ·−−→CC1+−−→BC ·−−→AA1+−→CA ·−−→BB1= 0.2.) Let AA1, BB1, and CC1be the altitudes of a triangle ABC.Denote by a, b, c the lengths of sides BC, CA and AB resp ectively.Also, denote by~ha,~hb,~hcthe altitude vectors−−→AA1,−−→BB1, and−−→CC1respectively. Prove thata2~ha+ b2~hb+ c2~hc=~0.(Hint: Write~hain terms of−→AB and the projection of−→AB onto−−→BC.Here you can use projection formula from the text. Do the same forother altitude vectors.)3.) Suppose that ABC D is a rectangle in R3. Let M be an arbitrarypoint in R3. Prove that−−→M A ·−−→M C =−−→M B ·−−→M D.Using this equality, prove thatk−−→M Ak2+ k−−→M Ck2= k−−→M Dk2+ k−−→M Bk2.4.) Suppose that ~a,~b, ~c are coplanar (i.e. lie in the same plane in R3).Prove that ~a +~b,~b + ~c, and ~c + ~a are also coplanar.5.) Suppose that A, B, and C are three points in R3with positionvectors ~r1, ~r2, and ~r3respectively. Prove that the area of the triangleABC is given by12k~r1× ~r2+ ~r2× ~r3+ ~r3× ~r1k.6.) Suppose that we are given two perpendicular vectors ~a and~bin R3. We are also given a scalar k. Find a vector ~r satisfying thefollowing system of equations:~a × ~r =~b; ~a · ~r = kk~ak2.(Hint: Apply ~a× from the left to both sides of the first equation, anduse formulas from exercise 64 (page 751).)127.) Prove that(~a ×~b) · (~c ×~d) + (~b × ~c) · (~a ×~d) + (~c × ~a) · (~b ×~d) = 0.(Hint: You may want to use the formula from exercise 60 on page 751together with exercise 64 on page