Exercise Zone : Proyeksi Vektor (Vector Projection)

Berikut ini adalah kumpulan soal mengenai Proyeksi Vektor tipe standar. Jika ingin bertanya soal, silahkan gabung ke grup Facebook atau Telegram.

No. 1

Jika proyeksi orthogonal vektor {\vec{u}=3\vec{i}+4\vec{j}} pada vektor {\vec{v}=-4\vec{i}+8\vec{j}} adalah vektor \vec{w}, maka \left|\vec{w}\right| adalah ....

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$\begin{array}{rl}\left|\overrightarrow{v}\right|& =\sqrt{(-4{)}^2+(8{)}^2}\\ & =\sqrt{16+64}\\ & =\sqrt{80}\\ & =4\sqrt{5}\end{array}$

$\begin{array}{rl}\left|\overrightarrow{w}\right|& ={\displaystyle \frac{\overrightarrow{u}\cdot \overrightarrow{v}}{\left|\overrightarrow{v}\right|}}\\ & ={\displaystyle \frac{\left(\begin{array}{c}3\\ 4\end{array}\right)\cdot \left(\begin{array}{c}-4\\ 8\end{array}\right)}{4\sqrt{5}}}\\ & ={\displaystyle \frac{-12+32}{4\sqrt{5}}}\\ & ={\displaystyle \frac{20}{4\sqrt{5}}}\\ & ={\displaystyle \frac{5}{\sqrt{5}}}\cdot {\displaystyle \frac{\sqrt{5}}{\sqrt{5}}}\\ & ={\displaystyle \frac{5\sqrt{5}}{5}}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle \sqrt{5}}}}}\end{array}$

Jadi, \left|\vec{w}\right|=\sqrt5.

No. 2

Vektor yang merupakan proyeksi vektor {\vec{i}-2\vec{j}+3\vec{k}} pada {2\vec{i}+\vec{j}+\vec{k}} adalah

1. {\vec{i}+\dfrac12\vec{j}+2\vec{k}}
2. {4\vec{i}+2\vec{j}+2\vec{k}}
   
3. {\vec{i}+\dfrac12\vec{j}+\dfrac12\vec{k}}
   

4. {\vec{i}+2\vec{j}+\vec{k}}
5. {\vec{i}+\vec{j}+\dfrac12\vec{k}}
   

Misal \vec{a}=i-2j+3k, \vec{b}=2i+j+k, dan \vec{c} adalah vektor proyeksi \vec{a} pada \vec{b}.

$\begin{array}{rl}\left|\overrightarrow{b}\right|& =\sqrt{2^2+1^2+1^2}\\ & =\sqrt{4+1+1}\\ & =\sqrt{6}\end{array}$

$\begin{array}{rl}\overrightarrow{c}& ={\displaystyle \frac{a\cdot b}{{\left|\overrightarrow{b}\right|}^2}}\overrightarrow{b}\\ & ={\displaystyle \frac{(i-2j+3k)\cdot (2i+j+k)}{{\left(\sqrt{6}\right)}^2}}(2i+j+k)\\ & ={\displaystyle \frac{2-2+3}{6}}(2i+j+k)\\ & ={\displaystyle \frac{3}{6}}(2i+j+k)\\ & ={\displaystyle \frac{1}{2}}(2i+j+k)\\ & =\boxed{{\displaystyle \boxed{{\displaystyle i+{\displaystyle \frac{1}{2}}j+{\displaystyle \frac{1}{2}}k}}}}\end{array}$

Jadi, vektor yang merupakan proyeksi vektor {\vec{i}-2\vec{j}+3\vec{k}} pada {2\vec{i}+\vec{j}+\vec{k}} adalah {\vec{i}+\dfrac12\vec{j}+\dfrac12\vec{k}}.
JAWAB: C

No. 3

Jika proyeksi {\vec{u}=(2,4)} pada {\vec{w}=(3,1)} sama dengan proyeksi {\vec{v}=(a,-2)} pada \vec{w}. maka nilai a yang memenuhi adalah

1. -2
2. -3
   
3. 2
   

4. 3
5. 4
   

CARA 1

$\begin{array}{rl}{\left|\overrightarrow{w}\right|}^2& =3^2+1^2\\ & =9+1\\ & =10\end{array}$

$\begin{array}{rl}{\displaystyle \frac{\overrightarrow{u}\cdot \overrightarrow{w}}{{\left|\overrightarrow{w}\right|}^2}}\overrightarrow{w}& ={\displaystyle \frac{\overrightarrow{v}\cdot \overrightarrow{w}}{{\left|\overrightarrow{w}\right|}^2}}\overrightarrow{w}\\ {\displaystyle \frac{\left(\begin{array}{c}2\\ 4\end{array}\right)\cdot \left(\begin{array}{c}3\\ 1\end{array}\right)}{10}}\left(\begin{array}{c}3\\ 1\end{array}\right)& ={\displaystyle \frac{\left(\begin{array}{c}a\\ -2\end{array}\right)\cdot \left(\begin{array}{c}3\\ 1\end{array}\right)}{10}}\left(\begin{array}{c}3\\ 1\end{array}\right)\\ {\displaystyle \frac{6+4}{10}}\left(\begin{array}{c}3\\ 1\end{array}\right)& ={\displaystyle \frac{3a-2}{10}}\left(\begin{array}{c}3\\ 1\end{array}\right)\\ {\displaystyle \frac{10}{10}}\left(\begin{array}{c}3\\ 1\end{array}\right)& =\left(\begin{array}{c}{\displaystyle \frac{9a-6}{10}}\\ {\displaystyle \frac{3a-2}{10}}\end{array}\right)\\ \left(\begin{array}{c}3\\ 1\end{array}\right)& =\left(\begin{array}{c}{\displaystyle \frac{9a-6}{10}}\\ {\displaystyle \frac{3a-2}{10}}\end{array}\right)\end{array}$

$\begin{array}{rl}1& ={\displaystyle \frac{3a-2}{10}}\\ 10& =3a-2\\ 12& =3a\\ a& ={\displaystyle \frac{12}{3}}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle 4}}}}\end{array}$

CARA CEPAT

$\begin{array}{rl}\overrightarrow{u}\cdot \overrightarrow{w}& =\overrightarrow{v}\cdot \overrightarrow{w}\\ \left(\begin{array}{c}2\\ 4\end{array}\right)\cdot \left(\begin{array}{c}3\\ 1\end{array}\right)& =\left(\begin{array}{c}a\\ -2\end{array}\right)\cdot \left(\begin{array}{c}3\\ 1\end{array}\right)\\ 6+4& =3a-2\\ 10& =3a-2\\ 12& =3a\\ a& ={\displaystyle \frac{12}{3}}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle 4}}}}\end{array}$

Jadi, a=4.
JAWAB: E

No. 4

Jika panjang proyeksi vektor (a, 5, -1) pada vektor (1, 4, 8) adalah 2, maka a =

1. 2
2. 3
3. 4

4. 5
5. 6

SKMP Juli 2020

$\begin{array}{rl}{\displaystyle \frac{(a)(1)+(5)(4)+(-1)(8)}{\sqrt{1^2+4^2+8^2}}}& =2\\ {\displaystyle \frac{a+20-8}{\sqrt{1+16+64}}}& =2\\ {\displaystyle \frac{a+12}{\sqrt{81}}}& =2\\ {\displaystyle \frac{a+12}{9}}& =2\\ a+12& =18\\ a& =\boxed{{\displaystyle \boxed{{\displaystyle 6}}}}\end{array}$

Jadi, a =6.
JAWAB: E