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Diketahui vektor-vektor sebagai berikut :

{:[u=[1","-2","3]],[v=[5","6","-1]],[w=[3","2","1]]:}

(15 poin) Apakah vektor
z=[0,16,-16] merupakan kombinasi linier dari vektorvektor
u,v, dan
w ?

Diketahui vektor-vektor sebagai berikut : $\newline$ $\begin{array}{l} \mathbf{u}=[1,-2,3] \\ \mathbf{v}=[5,6,-1] \\ \mathbf{w}=[3,2,1] \end{array}$ $\newline$ $1$ . ( $15$ poin) Apakah vektor $\mathbf{z}=[0,16,-16]$ merupakan kombinasi linier dari vektorvektor $\mathbf{u}, \mathbf{v}$ , dan $\mathbf{w}$ ?

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Solve quadratic equations: word problems

Full solution

Q. Diketahui vektor-vektor sebagai berikut :

\newline

\begin{array}{l} \mathbf{u}=[1,-2,3] \\ \mathbf{v}=[5,6,-1] \\ \mathbf{w}=[3,2,1] \end{array}

\newline

1

. (

15

poin) Apakah vektor

\mathbf{z}=[0,16,-16]

merupakan kombinasi linier dari vektorvektor

\mathbf{u}, \mathbf{v}

, dan

\mathbf{w}

?

Set up equations: To determine if $z$ is a linear combination of $u$ , $v$ , and $w$ , we need to solve the equation $au + bv + cw = z$ for scalars $a$ , $b$ , and $c$ .
Solve for $a$ : Set up the system of linear equations based on the components of the vectors: $\newline$ $1a + 5b + 3c = 0$ (x-components) $\newline$ $-2a + 6b + 2c = 16$ (y-components) $\newline$ $3a - 1b + 1c = -16$ (z-components)
Substitute $a$ into equations: Solve the first equation for $a$ : $a = -5b - 3c$ .
Simplify equations: Substitute $a$ into the second and third equations: $\newline$ $-2(-5b - 3c) + 6b + 2c = 16$ $\newline$ $3(-5b - 3c) - 1b + 1c = -16$
Combine like terms: Simplify the equations: $\newline$ $10b + 6c + 6b + 2c = 16$ $\newline$ $-15b - 9c - 1b + 1c = -16$
Solve system of equations: Combine like terms: $\newline$ $16b + 8c = 16$ $\newline$ $-16b - 8c = -16$
Solve system of equations: Combine like terms: $\newline$ $16b + 8c = 16$ $\newline$ $-16b - 8c = -16$ Solve the system of equations. Multiply the second equation by $-1$ to make it easier to add them together: $\newline$ $16b + 8c = 16$ $\newline$ $16b + 8c = 16$

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