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mathdefault\"}),\"u\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"accent-body\",\"style\":{\"left\":\"-0.20772em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"overlay\",\"style\":{\"height\":\"0.714em\",\"width\":\"0.471em\"}}),mdx(\"svg\",_extends({parentName:\"span\"},{\"width\":\"0.471em\",\"height\":\"0.714em\",\"style\":{\"width\":\"0.471em\"},\"viewBox\":\"0 0 471 714\",\"preserveAspectRatio\":\"xMinYMin\"}),mdx(\"path\",_extends({parentName:\"svg\"},{\"d\":\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\\nc-16-25.333-24-45-24-59z\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),\"\\u2223\")))),\" is just the x-coordinate of 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1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\\nc-16-25.333-24-45-24-59z\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"u\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t vlist-t2\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.151392em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-2.5500000000000003em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault mtight\"}),\"x\"))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}}),mdx(\"span\",{parentName:\"span\"}))))))))),\". Now, if we want to know the orthogonal projection of any vector in 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mathbb\"}),\"R\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"msupsub\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.8141079999999999em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"sizing reset-size6 size3 mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mtight\"}),\"2\")))))))))))),\" onto line \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"L\")),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"L\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.68333em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord mathdefault\"}),\"L\")))),\", we only need to find where 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mathdefault\"}),\"i\"))),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3.22852em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"accent-body\",\"style\":{\"left\":\"-0.2355em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"overlay\",\"style\":{\"height\":\"0.714em\",\"width\":\"0.471em\"}}),mdx(\"svg\",_extends({parentName:\"span\"},{\"width\":\"0.471em\",\"height\":\"0.714em\",\"style\":{\"width\":\"0.471em\"},\"viewBox\":\"0 0 471 714\",\"preserveAspectRatio\":\"xMinYMin\"}),mdx(\"path\",_extends({parentName:\"svg\"},{\"d\":\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\\nc-16-25.333-24-45-24-59z\"})))))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mpunct\"}),\",\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.16666666666666666em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mspace\"}),\"\\xA0\"),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord accent\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t 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4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\\nc-16-25.333-24-45-24-59z\"}))))))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-s\"}),\"\\u200B\")),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.19444em\"}}),mdx(\"span\",{parentName:\"span\"})))))))),\" land.\\nNote: The projection outputs a scalar, the orange dot on the number line, not a 2D vector lying on the line. What we get is the component of a vector in the direction \",mdx(\"span\",_extends({parentName:\"p\"},{\"className\":\"katex\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-mathml\"}),mdx(\"math\",_extends({parentName:\"span\"},{\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"}),mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mover\",_extends({parentName:\"mrow\"},{\"accent\":\"true\"}),mdx(\"mi\",{parentName:\"mover\"},\"u\"),mdx(\"mo\",{parentName:\"mover\"},\"\\u20D7\"))),mdx(\"annotation\",_extends({parentName:\"semantics\"},{\"encoding\":\"application/x-tex\"}),\"\\\\vec{u}\")))),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"katex-html\",\"aria-hidden\":\"true\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"base\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"strut\",\"style\":{\"height\":\"0.714em\",\"verticalAlign\":\"0em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord accent\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-t\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist-r\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"vlist\",\"style\":{\"height\":\"0.714em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"style\":{\"top\":\"-3em\"}}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"pstrut\",\"style\":{\"height\":\"3em\"}})),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord\"}),mdx(\"span\",_extends({parentName:\"span\"},{\"className\":\"mord 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