KI-generert løsningsforslag. Dette løsningsforslaget er skrevet av en språkmodell (Claude) og er ikke verifisert av lærer. Det kan inneholde regnefeil, upresis bruk av begreper eller mangler.a)
Vi beregner lengden av hver vektor med formelen ∣ v ⃗ ∣ = v 1 2 + v 2 2 |\vec{v}| = \sqrt{v_1^2 + v_2^2} ∣ v ∣ = v 1 2 + v 2 2
∣ a ⃗ ∣ = 2 2 + ( − 5 ) 2 = 4 + 25 = 29 ∣ b ⃗ ∣ = 1 2 + ( − 4 ) 2 = 1 + 16 = 17 ∣ c ⃗ ∣ = ( − 2 ) 2 + 10 2 = 4 + 100 = 104 ∣ d ⃗ ∣ = 4 2 + 1 2 = 16 + 1 = 17 \begin{aligned}
|\vec{a}| &= \sqrt{2^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29} \\
|\vec{b}| &= \sqrt{1^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} \\
|\vec{c}| &= \sqrt{(-2)^2 + 10^2} = \sqrt{4 + 100} = \sqrt{104} \\
|\vec{d}| &= \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17}
\end{aligned} ∣ a ∣ ∣ b ∣ ∣ c ∣ ∣ d ∣ = 2 2 + ( − 5 ) 2 = 4 + 25 = 29 = 1 2 + ( − 4 ) 2 = 1 + 16 = 17 = ( − 2 ) 2 + 1 0 2 = 4 + 100 = 104 = 4 2 + 1 2 = 16 + 1 = 17 b ⃗ \vec{b} b d ⃗ \vec{d} d ∣ b ⃗ ∣ = ∣ d ⃗ ∣ = 17 |\vec{b}| = |\vec{d}| = \sqrt{17} ∣ b ∣ = ∣ d ∣ = 17
b)
To vektorer er normalt på hverandre hvis og bare hvis prikkproduktet er null. Vi sjekker alle par:
a ⃗ ⋅ b ⃗ = 2 ⋅ 1 + ( − 5 ) ⋅ ( − 4 ) = 2 + 20 = 22 a ⃗ ⋅ c ⃗ = 2 ⋅ ( − 2 ) + ( − 5 ) ⋅ 10 = − 4 − 50 = − 54 a ⃗ ⋅ d ⃗ = 2 ⋅ 4 + ( − 5 ) ⋅ 1 = 8 − 5 = 3 b ⃗ ⋅ c ⃗ = 1 ⋅ ( − 2 ) + ( − 4 ) ⋅ 10 = − 2 − 40 = − 42 b ⃗ ⋅ d ⃗ = 1 ⋅ 4 + ( − 4 ) ⋅ 1 = 4 − 4 = 0 c ⃗ ⋅ d ⃗ = ( − 2 ) ⋅ 4 + 10 ⋅ 1 = − 8 + 10 = 2 \begin{aligned}
\vec{a} \cdot \vec{b} &= 2 \cdot 1 + (-5) \cdot (-4) = 2 + 20 = 22 \\
\vec{a} \cdot \vec{c} &= 2 \cdot (-2) + (-5) \cdot 10 = -4 - 50 = -54 \\
\vec{a} \cdot \vec{d} &= 2 \cdot 4 + (-5) \cdot 1 = 8 - 5 = 3 \\
\vec{b} \cdot \vec{c} &= 1 \cdot (-2) + (-4) \cdot 10 = -2 - 40 = -42 \\
\vec{b} \cdot \vec{d} &= 1 \cdot 4 + (-4) \cdot 1 = 4 - 4 = \textcolor{seagreen}{0} \\
\vec{c} \cdot \vec{d} &= (-2) \cdot 4 + 10 \cdot 1 = -8 + 10 = 2
\end{aligned} a ⋅ b a ⋅ c a ⋅ d b ⋅ c b ⋅ d c ⋅ d = 2 ⋅ 1 + ( − 5 ) ⋅ ( − 4 ) = 2 + 20 = 22 = 2 ⋅ ( − 2 ) + ( − 5 ) ⋅ 10 = − 4 − 50 = − 54 = 2 ⋅ 4 + ( − 5 ) ⋅ 1 = 8 − 5 = 3 = 1 ⋅ ( − 2 ) + ( − 4 ) ⋅ 10 = − 2 − 40 = − 42 = 1 ⋅ 4 + ( − 4 ) ⋅ 1 = 4 − 4 = 0 = ( − 2 ) ⋅ 4 + 10 ⋅ 1 = − 8 + 10 = 2 Kun b ⃗ ⋅ d ⃗ = 0 \vec{b} \cdot \vec{d} = 0 b ⋅ d = 0
b ⃗ \vec{b} b d ⃗ \vec{d} d
c)
To vektorer [ x 1 , y 1 ] [x_1, y_1] [ x 1 , y 1 ] [ x 2 , y 2 ] [x_2, y_2] [ x 2 , y 2 ] x 1 y 2 − x 2 y 1 = 0 x_1 y_2 - x_2 y_1 = 0 x 1 y 2 − x 2 y 1 = 0
a ⃗ , b ⃗ : 2 ⋅ ( − 4 ) − ( − 5 ) ⋅ 1 = − 8 + 5 = − 3 ≠ 0 a ⃗ , c ⃗ : 2 ⋅ 10 − ( − 5 ) ⋅ ( − 2 ) = 20 − 10 = 10 ≠ 0 a ⃗ , d ⃗ : 2 ⋅ 1 − ( − 5 ) ⋅ 4 = 2 + 20 = 22 ≠ 0 b ⃗ , c ⃗ : 1 ⋅ 10 − ( − 4 ) ⋅ ( − 2 ) = 10 − 8 = 2 ≠ 0 b ⃗ , d ⃗ : 1 ⋅ 1 − ( − 4 ) ⋅ 4 = 1 + 16 = 17 ≠ 0 c ⃗ , d ⃗ : ( − 2 ) ⋅ 1 − 10 ⋅ 4 = − 2 − 40 = − 42 ≠ 0 \begin{aligned}
\vec{a},\,\vec{b} &: \quad 2 \cdot (-4) - (-5) \cdot 1 = -8 + 5 = -3 \neq 0 \\
\vec{a},\,\vec{c} &: \quad 2 \cdot 10 - (-5) \cdot (-2) = 20 - 10 = 10 \neq 0 \\
\vec{a},\,\vec{d} &: \quad 2 \cdot 1 - (-5) \cdot 4 = 2 + 20 = 22 \neq 0 \\
\vec{b},\,\vec{c} &: \quad 1 \cdot 10 - (-4) \cdot (-2) = 10 - 8 = 2 \neq 0 \\
\vec{b},\,\vec{d} &: \quad 1 \cdot 1 - (-4) \cdot 4 = 1 + 16 = 17 \neq 0 \\
\vec{c},\,\vec{d} &: \quad (-2) \cdot 1 - 10 \cdot 4 = -2 - 40 = -42 \neq 0
\end{aligned} a , b a , c a , d b , c b , d c , d : 2 ⋅ ( − 4 ) − ( − 5 ) ⋅ 1 = − 8 + 5 = − 3 = 0 : 2 ⋅ 10 − ( − 5 ) ⋅ ( − 2 ) = 20 − 10 = 10 = 0 : 2 ⋅ 1 − ( − 5 ) ⋅ 4 = 2 + 20 = 22 = 0 : 1 ⋅ 10 − ( − 4 ) ⋅ ( − 2 ) = 10 − 8 = 2 = 0 : 1 ⋅ 1 − ( − 4 ) ⋅ 4 = 1 + 16 = 17 = 0 : ( − 2 ) ⋅ 1 − 10 ⋅ 4 = − 2 − 40 = − 42 = 0 Ingen av vektorene er parallelle.