Let a1=[13−1],a2=[−6−143],b=[53h]. for what value(s) of h is b in the plane spanned by a1 and a2

b in the plane spanned by a1 and a2 if b is the linear combination of a1 and a2. b=θ1a1+θ2a2 ⇒[53h]=α1[13−1],α2=[−6−143] ⇒5=α1−6α2−−−1 3=3α1−14α2−−−2 h=−α1−3α2−−−3 1×3⇒15=3α1−18α2 2⇒3=3α1−14α2 Subtract, 12=−14α2⇒α2=−3 Sub α2=−3 in 1, 5=α1+18 ⇒α1=−13 Sub α1=−13 and α2=−3 in 3, h=13+3(−3)=13−9=4 i.e h=4

Check the solution to "Let a1=[13−1],a2=[−6−143],b=[53h]. for what value(s) of h is..."

College
Algebra
Linear algebra
Matrix transformations

William Boggs

Answered question

2021-12-13

Let

a_{1} = [\begin{matrix} 1 \\ 3 \\ - 1 \end{matrix}], a_{2} = [\begin{matrix} - 6 \\ - 14 \\ 3 \end{matrix}], b = [\begin{matrix} 5 \\ 3 \\ h \end{matrix}]

. for what value(s) of h is b in the plane spanned by

a_{1}

and

a_{2}

Answer & Explanation

Ana Robertson

Beginner2021-12-14Added 26 answers

b in the plane spanned by

a_{1}

and

a_{2}

if b is the linear combination of

a_{1}

and

a_{2}

b = θ_{1} a_{1} + θ_{2} a_{2}

\Rightarrow [\begin{matrix} 5 \\ 3 \\ h \end{matrix}] = α_{1} [\begin{matrix} 1 \\ 3 \\ - 1 \end{matrix}], α_{2} = [\begin{matrix} - 6 \\ - 14 \\ 3 \end{matrix}]

\Rightarrow 5 = α_{1} - 6 α_{2} - - - 1

3 = 3 α_{1} - 14 α_{2} - - - 2

h = - α_{1} - 3 α_{2} - - - 3

1 \times 3 \Rightarrow 15 = 3 α_{1} - 18 α_{2}

2 \Rightarrow 3 = 3 α_{1} - 14 α_{2}

Subtract,

12 = - 14 α_{2} \Rightarrow α_{2} = - 3

Sub

α_{2} = - 3

in 1,

5 = α_{1} + 18

\Rightarrow α_{1} = - 13

Sub

α_{1} = - 13

and

α_{2} = - 3

in 3,

h = 13 + 3 (- 3) = 13 - 9 = 4

i.e

h = 4

Paul Mitchell

Beginner2021-12-15Added 40 answers

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