■♥✲❈❧❛ss ❚❡st ★✶ ✲ ▼♦❞✉❧❡ ❱❡❝t♦r ❈❛❧❝✉❧✉s
                       ❊♥❣✐♥❡❡r✐♥❣ ▼❛t❤❡♠❛t✐❝s ❢♦r ❆❞✈❛♥❝❡❞ ❙t✉❞✐❡s
                       ■■❚ ❉❤❛r✇❛❞
                       ❆✉t✉♠♥ ✷✵✶✾
                       ❚✐♠❡ ✲ ✷✺ ♠✐♥✉t❡s
                       ▼❛①✐♠✉♠ s❝♦r❡ ✲ ✷✵
                       ❘✉❧❡ ❢♦r ❛❜s❡♥t❡❡ ✲ ▼✐♥✐♠✉♠ ✸✵✪ ♣❡♥❛❧t②✱ ❞✐s❝✉ss r❡❛s♦♥s ❛❜s❡♥s❡ ✐♥ ♣❡rs♦♥ t♦ ❣❡t ❛
                   ❝❤❛♥❝❡ ❢♦r r❡✲t❡st✳
                       ◆♦t❡✿
                      ✶✳ ❈♦♥s✐❞❡r✐♥❣ t❤❡ t✐♠❡ ❛❧❧♦tt❡❞ ❢♦r t❤❡ q✉✐③✱ t♦ ❜❡ ♦♥ t❤❡ s❛❢❡r s✐❞❡✱ st✉❞❡♥t ♠❛② ✇❛♥t
                         t♦ ✜rst ❣♦ t❤r♦✉❣❤ ❛❧❧ q✉❡st✐♦♥s ❛♥❞ ✐❞❡♥t✐❢② t❤❡ ♦♥❡s t❤❛t ✇♦✉❧❞ ♥♦t t❛❦❡ ♠✉❝❤ t✐♠❡
                         t♦ ❛♥s✇❡r✳ ❆♥s✇❡r t❤❡s❡ ✜rst ❛♥❞ t❤❡♥ ✇♦r❦ ♦♥ t❤❡ r❡❧❛t✐✈❡❧② t❡❞✐♦✉s q✉❡st✐♦♥s✳
                      ✷✳ ❊♥s✉r❡ t♦ ❝❧❡❛r❧② ✇r✐t❡ q✉❡st✐♦♥ ♥✉♠❜❡r ❜❡❢♦r❡ t❤❡ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✳
                      ✸✳ ❲♦r❦❡❞ ♦✉t s♦❧✉t✐♦♥s ♦♥ s✉♣♣❧✐♠❡♥ts ❛r❡ ♠✉st ❢♦r s♦♠❡ ♣r♦❜❧❡♠s✳
                      ✹✳ P❧❡❛s❡ ❡♥s✉r❡ t♦ ✇r✐t❡ ◗✉❡st✐♦♥ ♥✉♠❜❡r ✐♥ ❛ ❜♦① ❛s ❛ ❤❡❛❞✐♥❣ t♦ t❤❡ ✉♣❝♦♠✐♥❣
                         ❛♥s✇❡r ♦♥ s✉♣♣❧✐♠❡♥ts ❡✳❣✳
                      ✺✳  ◗✉❡st✐♦♥ ✶
                                                           3
                      ✶✳ ■❢ u,v,w ❛r❡ ✈❡❝t♦rs ✐♥ R ❛♥❞ k ❛♥❞ l ❛r❡ s❝❛❧❛rs st❛t❡ ❚r✉❡✴❋❛❧s❡✿                                               ✭♠❛r❦ ✷✮
                          ✭❛✮ u × v = v × u             ❴❴❴❴❴❴❴❴❴❴❴
                          ✭❜✮ u · (v × w) = v.(u × w)                ❴❴❴❴❴❴❴❴❴❴❴
                           ✭❝✮ w · (v × u) = −w.(u × u)                ❴❴❴❴❴❴❴❴❴❴❴
                          ✭❞✮ w · (v × u) = v.(u × w)                ❴❴❴❴❴❴❴❴❴❴❴
                           ✭❡✮ kv + wk = kvk+kwk                 ❴❴❴❴❴❴❴❴❴❴❴
                           ✭❢✮ u · v = v · u
                      ✷✳ ❚✇♦ s♦❛♣ ❜✉❜❜❧❡s ❛r❡ ❝♦❛❧❡s❝✐♥❣ t♦❣❡t❤❡r✳ ❖♥❡ ✐s ❤❛✈✐♥❣ r❛❞✐✉s ♦❢ ✸❝♠ ✇✐t❤ ❝❡♥t❡r
                         ❛t ✭✵✱✵✱✵✮ ❛♥❞ ♦t❤❡r ❤❛✈✐♥❣ r❛❞✐✉s ♦❢ ✷❝♠ ❛♥❞ ❝❡♥t❡r ✭✷✱✸✱✶✮✳ ■❢ r = ui + vj + wk
                         ❞❡♥♦t❡s ❧♦❝✉s ♦❢ ❛❧❧ t❤❡ ♣♦✐♥ts ✐♥ ✸❉ s♣❛❝❡ ❝❛♥ ②♦✉ ♣r♦✈✐❞❡ ✇❤❛t ✐s ❡①♣❡❝t❡❞ t②♣❡
                         ♦❢ ❣❡♦♠❡tr② ♦❢ t❤❛t s❤❛r❡❞ ❧♦❝✐✐❄ ❈❛♥ ②♦✉ ✐❞❡♥t✐❢② u(x,y,z)✱ v(x,y,z)✱ ❛♥❞ w(x,y,z)❄                              ✭♠❛r❦ ✸✮
                      ✸✳ ❲r✐t❡ t❤❡ ✈❡❝t♦r ❢♦r♠ ♦❢ ❛ ❧✐♥❡ ▲ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t P = (2,3,5) ❛♥❞ ♣❛r❛❧❧❡❧ t♦ t❤❡
                         ✈❡❝t♦r v = (4,−1,6)                                                                                               ✭♠❛r❦ ✷✮
                      ✹✳ ■❢ ❛ ❢❛r♠ ♣r♦❞✉❝❡s ✶✵ ✉♥✐ts ♦❢ ♣r♦❞✉❝❡ ♣❡r ✉♥✐t ❛r❡❛✳ ❍♦✇ ♠✉❝❤ ❝r♦♣ ♣r♦❞✉❝❡ ✐s ❡①✲
                         ♣❡❝t❡❞❢r♦♠❛❢❛r♠✇❤♦s❡❢♦✉r❝♦r♥❡r♣♦✐♥ts❛r❡❣✐✈❡♥❛s✲A(2,2),B(6,3),C(7,5),D(4,6)
                                                                                                                                           ✭♠❛r❦ ✹✮
                                                           ˆ             ˆ         ˆ
                      ✺✳ ❋♦r ❛ ❝✉r✈❡ r(t) = acos(t)i + bsin(t)j + c(t)k                                                                    ✭♠❛r❦ ✻✮
                          ✭❛✮ ❲❤❛t ✐s t❤❡ ❣❡♦♠❡tr② ❡①♣❡❝t❡❞❄
                          ✭❜✮ ❲❤❛t ✐s t❤❡ ✉♥✐t ✈❡❝t♦r ❛❧♦♥❣ t❛♥❣❡♥t ❞✐r❡❝t✐♦♥ ❛t t❂✷
                           ✭❝✮ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ tr❛✈❡rs❡❞ ❢r♦♠ t❂✶ t♦ t❂✷ ❛❧♦♥❣ t❤❡ ❝✉r✈❡
                                                                             ✶
                   ✻✳ ■❢ ✈❡❝t♦r p s✉❝❤ t❤❛t ✱ p = r′(t) ✇❤❡r❡ r ✐s ♣♦s✐t✐♦♥ ✈❡❝t♦r ✇❤✐❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s
                      ❛♣♣r♦♣r✐❛t❡ ❢♦r ❛♥② ✈❡❝t♦r a
                      tr✉❡✴❢❛❧s❡✿                                                                                     ✭♠❛r❦ ✸✮
                       ✭❛✮ ❛✳♣ ✐s ❝♦♠♣♦♥❡♥t ❛❧♦♥❣ ♣          ❴❴❴❴❴❴❴❴❴❴❴
                      ✭❜✮ p(t) ✐s ❛ ✉♥✐t ✈❡❝t♦r ❛❧✇❛②s        ❴❴❴❴❴❴❴❴❴❴❴
                       ✭❝✮ ✐❢ q(t) ✐s ✉♥✐t ✈❡❝t♦r ❛❧♦♥❣ p(t), q · q′ = 0      ❴❴❴❴❴❴❴❴❴❴❴
                                                                 ✷