verkefnablad01_lausnir - Línuleg algebra A og B (STÆ107G,...

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Unformatted text preview: Línuleg algebra A og B (STÆ107G, STÆ106G, HAG105G) Verkefnablað 1 Lausnir 7.9.2011 1. Finnið pólhnit vigursins v ef: (a) v = (3 , 4) (b) v = (- 1 , 5) (c) v = (2 ,- 4) (d) v = (- 2 ,- 3) Lausn: (a) Lengd vigursins er | v | = √ 3 2 + 4 2 = 5 og stefnuhorn hans er θ = arctan( 4 3 ) ≈ . 927295 . (b) Lengd vigursins er | v | = p (- 1) 2 + 5 2 = √ 26 og stefnuhorn hans er θ = arctan( 1 5 )+ π 2 ≈ 1 . 768192 . (c) Lengd vigursins er | v | = p 2 2 + (- 4) 2 = 2 √ 5 og stefnuhorn hans er θ = arctan(- 4 2 ) ≈- 1 . 107149 . (d) Lengd vigursins er | v | = p (- 2) 2 + (- 3) 2 = √ 13 og stefnuhorn hans er θ = arctan( 3 2 ) + π ≈ 4 . 124386 . 2. Höfum vigrana a = (1 , 3) , b = (- 1 , 2) og c = (2 ,- 4) . Teiknið eftirfarandi vigra í hnitaker : (a) a + 2 b (b) a- 3( b + c ) (c) 2 · c- a- . 5 b Lausn: 3. Tökum sömu vigra og í dæminu á undan. Reiknið eftirfarandi: (a) 2 a · c (b) a · ( c + b ) (c) Hornið milli vigranna a og b Lausn: (a) 2 a · c = 2(1 , 3) · (2 ,- 4) = (2 , 6) · (2 ,- 4) = 2 · 2 + 6 · (- 4) =- 20 . (b) a · ( c + b ) = (1 , 3) · ((2 ,- 4) + (- 1 , 2)) = (1 , 3) · (1 ,- 2) = 1 · 1 + 3 · (- 2) =- 5 . (c) Ef θ er hornið milli a og b þá höfum við að cos( θ ) = a · b | a || b | = (1 , 3) · (- 1 , 2) √ 1 2 +3 2 √ (- 1) 2 +2 2 = 5 5 √ 2 = 1 √ 2 ....
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This note was uploaded on 10/25/2011 for the course VéLAVERKF STÆ107G taught by Professor Ragnarsigurðsson during the Fall '11 term at Uni. Iceland.

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verkefnablad01_lausnir - Línuleg algebra A og B (STÆ107G,...

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