Respuesta :
[tex] \bf \begin{cases} u=<-7,6>\\ v=<-4,17> \end{cases}~\hspace{7em}3v\implies 3<-4,17>\implies <-12,51> \\\\\\ u+3v\implies <-7,6>+<-12,51>\implies <-7-12,6+51> \\\\\\ <-19,57> \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ <-19,57>+<x,y>=<1,0>\implies \begin{cases} -19+x=1\implies &x=20\\ 57+y=0\implies &y=-57 \end{cases} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill <20,-57>~\hfill [/tex]
The value of the vector w is <20, -57>.
Vector Addition
Whenever we add two vectors such that [tex]\bold{<x_1,\ y_1>}[/tex] and [tex]\bold{<x_2,\ y_2>}[/tex], we add the corresponding component of one vector to the corresponding component of another vector, and the result vector is known as the resultant vector.
Therefore, [tex]\bold{<x_1,\ y_1>+<x_1,\ y_1>=<(x_1+x_2),\ (y_1+y_2)>}[/tex]
Vector Multiplication
Whenever we multiply a vector such as <x, y> with a whole number a, then each component of the vector is multiplied by the whole number. therefore,
[tex]\bold{a \times <x,\ y>= <ax,\ ay>}[/tex]
For Vector U, V, and W
Given to us,
u = <-7, 6>,
v = <-4, 17>,
resultant, R = <1, 0>
We can write it as,
[tex]\bold{\underset{u}{\rightarrow}+\underset{3v}{\rightarrow}+\underset{w}{\rightarrow}\ =\ \underset{R}{\rightarrow}}[/tex]
substituting the values,
[tex]\bold{<-7,\ 6>+3<-4,\ 17>+<x_w,\ y_w>=<1,\ 0>}[/tex]
[tex]\bold{<-7,\ 6>+<-12,\ 51>+<x_w,\ y_w>=<1,\ 0>}[/tex]
[tex]\bold{<-19,\ 57>+<x_w,\ y_w>=<1,\ 0>}[/tex]
[tex]\bold{<x_w,\ y_w>=<1,\ 0>-<-19,\ 57>}[/tex]
[tex]\bold{<x_w,\ y_w>=<20,\ -57>}[/tex]
Hence, the value of the vector w is <20, -57>.
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