Proyeksi merupakan ilmu yang mempelajari tentang cara pandang objek dalam ruang dimensi tiga dalam gambar di ruang dimensi dua. Cara ini mempermudah kita untuk melihat objek yang terletak di ruang dimensi tiga. Pada proyeksi vektor, objek yang diproyeksikan berupa vektor, baik itu panjangnya atau vektor itu sendiri. Proyeksi dibedakan menjadi beberapa jenis, di antaranya adalah proyeksi ortogonal, aksonometri, proyeksi miring (oblique), dan perspektif. Pada pembahasan proyeksi vektor kali ini hanya akan membahas mengenai proyeksi vektor ortogonal. Jadi, untuk jenis proyeksi lainnya tidak akan dibahas pada halaman ini.
alt="" src="http://idschool.net/wp-content/uploads/2017/11/Hasil-Proyeksi-Skalar-dan-Vektor-Ortogonal-e1510984758329.png" style="height:200px; width:234px" />
Proyeksi ortogonal adalah cara pandang mata pada sebuah objek yang ditarik garis tegak lurus pada sebuah bidang datar. Terdapat dua proyeksi ortogonal yang akan di bahas pada pembahasan kali ini, yaitu proyeksi skalar dan vektor ortogonal. Perhatikan gambar dua proyeksi vektor dengan arah yang berbeda pada gambar di bawah
alt="Proyeksi Skalar dan Proyeksi Vektor Ortogonal" src="http://idschool.net/wp-content/uploads/2017/11/Proyeksi-Skalar-dan-Vektor-Orthogonal-e1510981459204.png" style="height:200px; width:371px" />
Proyeksi Skalar Ortogonal
Proyeksi skalar ortogonal biasa disebut juga dengan proyeksi panjang vektor ortogonal. Dalam kata lainnya, objek proyeksi adalah panjang vektor. Rumus untuk menghitung panjang proyeksi skalar vektor ortogonal adalah sebagai berikut.
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Proyeksi skalar ortogonal alt="\vec{a}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-85a7117c3b11a1fe91637dfe11762336_l3.svg" style="height:11px; width:9px" title="Rendered by QuickLaTeX.com" /> pada arah vektor alt="\vec{b}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-cd7c50e7e8a69aec1cf47b04509a672f_l3.svg" style="height:15px; width:8px" title="Rendered by QuickLaTeX.com" />.
alt="\[ \left| \vec{c} \right| = \frac{\vec{a} \cdot \vec{b}}{ \left| \vec{b} \right| } \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-5f538d8bb5515b7413e42160cb3237b3_l3.svg" style="height:50px; width:61px" title="Rendered by QuickLaTeX.com" />
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Proyeksi skalar ortogonal alt="\vec{b}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-cd7c50e7e8a69aec1cf47b04509a672f_l3.svg" style="height:15px; width:8px" title="Rendered by QuickLaTeX.com" /> pada arah vektor alt="\vec{a}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-85a7117c3b11a1fe91637dfe11762336_l3.svg" style="height:11px; width:9px" title="Rendered by QuickLaTeX.com" />.
alt="\[ \left| \vec{c} \right| = \frac{\vec{a} \cdot \vec{b}}{ \left| \vec{a} \right| } \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-4e62195355675c71de868a8e7e49b426_l3.svg" style="height:39px; width:61px" title="Rendered by QuickLaTeX.com" />
Proyeksi Vektor Ortogonal
Objek pada proyeksi skalar vektor ortogonal adalah panjang proyeksi vektor. Sedangkan pada proyeksi vektor ortogonal yang menjadi objek utamanya adalah vektornya. Vektor hasil proyeksi dapat ditentukan melalui rumus berikut.
Contoh Soal dan Pembahasan
Panjang proyeksi ortogonal vektor alt="\vec{a} = (p, 2, 4)" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-511262542f7690d442f52243d5e54ea4_l3.svg" style="height:16px; width:72px" title="Rendered by QuickLaTeX.com" /> pada alt="\vec{b} = (2, p, 1)" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-4cee518fa33734904e75be4ed13fb871_l3.svg" style="height:19px; width:71px" title="Rendered by QuickLaTeX.com" /> adalah 4. Nilai p adalah ….
alt="\[ \textrm{A.} \; \; \; -4 \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-4d13da92780b632d1df27a01996031c4_l3.svg" style="height:12px; width:52px" title="Rendered by QuickLaTeX.com" />
alt="\[ \textrm{B.} \; \; \; -2 \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-55ccd38823d83e252ee3d69a8f8c5d98_l3.svg" style="height:11px; width:51px" title="Rendered by QuickLaTeX.com" />
alt="\[ \textrm{C.} \; \; \; - \frac{1}{2} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-c476fa2355dfb47007fe3a2a6b2c3fb7_l3.svg" style="height:31px; width:54px" title="Rendered by QuickLaTeX.com" />
alt="\[ \textrm{D.} \; \; \; \frac{1}{2} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-86591ec721014ed8a4c76cf3ef25f7b3_l3.svg" style="height:31px; width:37px" title="Rendered by QuickLaTeX.com" />
alt="\[ \textrm{E.} \; \; \; 2 \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-9b2797bbf22fe1c3da44634910055a4a_l3.svg" style="height:11px; width:33px" title="Rendered by QuickLaTeX.com" />
Pembahasan:
Mencari panjang vektor b:
alt="\[ \left| \vec{b} \right| = \sqrt{2^{2} + p ^{2} + 1^{2}} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-021020830631b16bed28320000f1f2c6_l3.svg" style="height:27px; width:125px" title="Rendered by QuickLaTeX.com" />
alt="\[ \left| \vec{b} \right| = \sqrt{4+ p ^{2} + 1} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-0b9ae4e2887dd57c668d1d02f5daa42f_l3.svg" style="height:27px; width:112px" title="Rendered by QuickLaTeX.com" />
alt="\[ \left| \vec{b} \right| = \sqrt{p ^{2} + 5} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-73f490775889c80d1aaef295760cdd4b_l3.svg" style="height:27px; width:86px" title="Rendered by QuickLaTeX.com" />
Beradasrkan rumus proyeksi skalar (proyeksi panjang) ortogonal vektor dapat diperoleh persamaan berikut.
alt="\[ \left| \vec{c} \right| = \frac{\vec{a} \cdot \vec{b}}{\left| \vec{b} \right|} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-c318a7163a6168c7868079a26e46bd9c_l3.svg" style="height:50px; width:61px" title="Rendered by QuickLaTeX.com" />
alt="\[ 4 = \frac{(p, 2, 4)(2, p, 1)}{\sqrt{p^{2}+5}} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-5ef99ae48953b7aee33d59a9a8151c7e_l3.svg" style="height:38px; width:121px" title="Rendered by QuickLaTeX.com" />
alt="\[ 4 = \frac{2p + 2p + 4}{\sqrt{p^{2}+5}} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-437c5d33ab609e4384fa3febbd331a6e_l3.svg" style="height:38px; width:101px" title="Rendered by QuickLaTeX.com" />
alt="\[ 4 = \frac{4p + 4}{\sqrt{p^{2}+5}} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-206680ab542b02b541dfdb21188a614f_l3.svg" style="height:38px; width:83px" title="Rendered by QuickLaTeX.com" />
alt="\[ 4 = \frac{4(p + 1)}{\sqrt{p^{2}+5}} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-f557d28d1291471de3a3eeff34084345_l3.svg" style="height:38px; width:83px" title="Rendered by QuickLaTeX.com" />
alt="\[ 1 = \frac{p + 1}{\sqrt{p^{2}+5}} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-3acb6b006e3dbfb1bf9f05a7db5c37c8_l3.svg" style="height:38px; width:82px" title="Rendered by QuickLaTeX.com" />
alt="\[ \sqrt{p^{2}+5} = p + 1 \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-cf353402cb1e57754f3e8961cdf4e5a2_l3.svg" style="height:18px; width:104px" title="Rendered by QuickLaTeX.com" />
alt="\[ p^{2}+5 = (p + 1)^{2} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-c7ee24878336d34f1fea45f0b64b9750_l3.svg" style="height:18px; width:109px" title="Rendered by QuickLaTeX.com" />
alt="\[ p^{2}+5 = p^{2} + 2p + 1 \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-f5b2a07053848abbe02370dfbcb72c2c_l3.svg" style="height:17px; width:130px" title="Rendered by QuickLaTeX.com" />
alt="\[ 5 = 2p + 1 \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-c38a96a31848219f8549de3f08045734_l3.svg" style="height:14px; width:66px" title="Rendered by QuickLaTeX.com" />
alt="\[ 2p = 5 - 1 \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-bfbfed547236a9a2823dab284bc32490_l3.svg" style="height:14px; width:66px" title="Rendered by QuickLaTeX.com" />
alt="\[ 2p = 4 \rightarrow p=\frac{4}{2} = 2 \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-85491c68a1af8469fd671b73b401977d_l3.svg" style="height:31px; width:127px" title="Rendered by QuickLaTeX.com" />
Jawaban: E
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Proyeksi vektor ortogonal alt="\vec{a}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-85a7117c3b11a1fe91637dfe11762336_l3.svg" style="height:11px; width:9px" title="Rendered by QuickLaTeX.com" /> pada alt="\vec{b}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-cd7c50e7e8a69aec1cf47b04509a672f_l3.svg" style="height:15px; width:8px" title="Rendered by QuickLaTeX.com" />.
alt="\[ \vec{c} = \frac{\vec{a} \cdot \vec{b} }{\left| \vec{b} \right| ^{2} } \cdot \vec{b} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-6ae2220cf89e707dc2c55744102bc436_l3.svg" style="height:53px; width:72px" title="Rendered by QuickLaTeX.com" />
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Proyeksi vektor ortogonal alt="\vec{b}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-cd7c50e7e8a69aec1cf47b04509a672f_l3.svg" style="height:15px; width:8px" title="Rendered by QuickLaTeX.com" /> pada alt="\vec{a}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-85a7117c3b11a1fe91637dfe11762336_l3.svg" style="height:11px; width:9px" title="Rendered by QuickLaTeX.com" />.
alt="\[ \vec{c} = \frac{\vec{a} \cdot \vec{b} }{\left| \vec{a} \right| ^{2} } \cdot \vec{a} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-e2731e7482c80af4bfc877f86bec0043_l3.svg" style="height:41px; width:73px" title="Rendered by QuickLaTeX.com" />