mohon bantuannya kak ( soal yang dilingkari).... Question from @cahyarufika

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Vektor

Soal 2.a.

$\begin{aligned}&\vec{u}=\begin{pmatrix}-6\\-3\\6\end{pmatrix}\ ,\quad\vec{v}=\begin{pmatrix}1\\1\\-2\end{pmatrix}\\\\&\textsf{Panjang vektor $\vec{u}$:}\\&\|\vec{u}\|=\sqrt{(-6)^2+(-3)^2+6^2}\\&{\quad\ \:}=\sqrt{36+9+36}\ =\ \sqrt{81}\\&{\quad\ \:}=\bf9\ satuan\ panjang\\\\&\textsf{Panjang vektor $\vec{v}$:}\\&\|\vec{v}\|=\sqrt{1^2+1^2+(-2)^2}\\&{\quad\ \:}=\sqrt{1+1+4}\\&{\quad\ \:}=\bf\sqrt{6}\ satuan\ panjang\\\\\end{aligned}$

Soal 2.b.

$\begin{aligned}2\vec{u}+9\vec{v}&=2\begin{pmatrix}-6\\-3\\6\end{pmatrix}+9\begin{pmatrix}1\\1\\-2\end{pmatrix}\\&=\begin{pmatrix}-12+9\\-6+9\\12+(-18)\end{pmatrix}\\&=\begin{pmatrix}\bf-3\\\bf3\\ \bf-6\end{pmatrix}\end{aligned}$

Panjang $2\vec{u}+9\vec{v}$ :

$\begin{aligned}\|2\vec{u}+9\vec{v}\|&=\sqrt{(-3)^2+3^2+6^2}\\&=\sqrt{9+9+36}\\&=\sqrt{9\cdot6}\\&=\bf3\sqrt{6}\ satuan\ panjang\end{aligned}$

Soal 4. a.

$\begin{aligned}&\vec{u}=\hat{i}+a\hat{j}\,,\quad\vec{v}=-4\hat{i}-2\hat{j}\\&\vec{w}=5\vec{u}+2\vec{v}\,,\quad\|\vec{w}\|=5\\\\&\sf Maka:\\&\vec{w}=5\left(\hat{i}+a\hat{j}\right)+2\left(-4\hat{i}-2\hat{j}\right)\\&{\;\;\,}=(5-8)\hat{i}+(5a-4)\hat{j}\\&{\;\;\,}=-3\hat{i}+(5a-4)\hat{j}\\\\&5=\sqrt{(-3)^2+(5a-4)^2}\\&25=(-3)^2+(5a-4)^2\\&{\ \ \;}=9+25a^2-40a+16\\&{\ \ \;}=25+25a^2-40a\\&0=25a^2-40a\\&\Leftrightarrow\ 0=5a(5a-8)\\&\Leftrightarrow\ a=\bf0\ {\sf atau}\ a=\bf\frac{8}{5}\end{aligned}$

∴ Jadi, nilai a adalah 0 atau 8/5.

Soal 4. b.

Karena ada dua kemungkinan nilai a, maka ada dua kemungkinan $\vec{w}$ .

Oleh karena itu, ada dua kemungkinan vektor satuan yang searah dengan $\vec{w}$ .

Jika a = 0, maka $\vec{w} = 3\hat{i}-4\hat{j}$ .
Vektor satuan yang searah dengan vektor $\vec{w}$ adalah:
$\begin{aligned}\hat{e}_{\vec{w}}&=\frac{1}{\|\vec{w}\|}\cdot\vec{w}\\&=\frac{1}{5}\left(3\hat{i}-4\hat{j}\right)\\\\\hat{e}_{\vec{w}}&=\bf\frac{3}{5}\,\hat{i}\ -\ \frac{4}{5}\,\hat{j}\end{aligned}$

Jika a = 8/5, maka $\vec{w} = 3\hat{i}+4\hat{j}$ .
Vektor satuan yang searah dengan vektor $\vec{w}$ adalah:
$\begin{aligned}\hat{e}_{\vec{w}}&=\frac{1}{\|\vec{w}\|}\cdot\vec{w}\\&=\frac{1}{5}\left(3\hat{i}+4\hat{j}\right)\\\\\hat{e}_{\vec{w}}&=\bf\frac{3}{5}\,\hat{i}\ +\ \frac{4}{5}\,\hat{j}\end{aligned}$

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