tutorial 7

# tutorial 7 - 6 Therefore ~ y-ˆ ~ y ="-6 3 ⇒ || ~...

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MATH1104C — Tutorial 7 1. Let ~u = 3 - 1 - 2 and ~x = 1 - 2 0 . 1) What is the orthogonal projection of ~u onto ~x ? 2) What is the orthogonal projection of ~x onto ~u ? Solution: 1) ˆ ~u = ~u ~x ~x ~x ~x = 5 5 ~x = ~x. 2) ˆ ~x = ~x ~u ~u ~u ~u = 5 14 ~u. 2. Let ~u = 1 1 1 ,~v = 1 - 1 0 , ~w = - 1 - 1 2 . 1) Show that the set of vectors { ~u,~v, ~w } is orthogonal. 2) Express ~x = 1 2 3 as a linear combination of ~u , ~v and ~w . Solution: 1) ~u ~v = ~u ~w = ~v ~w = 0. 2) Let ~x = c 1 ~u + c 2 ~v + c 3 ~w . Then c 1 = ~x ~u ~u ~u = 6 3 = 2 , c 2 = ~x ~v ~v ~v = - 1 2 , c 3 = ~x ~w ~w ~w = 3 6 = 1 2 . 1

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3. Let ~ y = " - 3 9 # , ~u = " 1 2 # . Find the distance from ~ y to the line through ~u and origin. Solution: ˆ ~ y = ~ y · ~u ~u · ~u ~u = 3 ~u = " 3
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Unformatted text preview: 6 # . Therefore ~ y-ˆ ~ y = "-6 3 # , ⇒ || ~ y-ˆ ~ y || = p (-6) 2 + 3 2 = √ 45 = 3 √ 5 . 4. Let W = span ~u = 1 1 ,~v = -1 1 , ~x = 3 2 1 . 1) Show that ~u ⊥ ~v . 2) Find proj W ~x . 5. Let ~u = 2 ,~v = 1 1 . Show that the angle between them is π 4 . 2...
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tutorial 7 - 6 Therefore ~ y-ˆ ~ y ="-6 3 ⇒ || ~...

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