pairs-of-lines-in-3d

Coincident & parallel lines

How do I tell if two lines are parallel?

• Two lines are parallel if, and only if, their direction vectors are parallel

  • This means the direction vectors will be scalar multiples of each other

  • For example, the lines whose equations are and <img alt=”begin mathsize 16px style bold r equals open parentheses table row 1 row cell negative 1 end cell row 5 end table close parentheses plus lambda subscript 2 open parentheses table row cell negative 1 end cell row 0 row 4 end table close parentheses end style” data-mathml='<math ><semantics><mstyle mathsize=”16px”><mi mathvariant=”bold”>r</mi><mo>=</mo><mfenced separators=”|”><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>+</mo><msub><mi>λ</mi><mn>2</mn></msub><mfenced separators=”|”><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></mstyle><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″}</annotation></semantics></math>’ height=”69″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2269%22%20width%3D%22136%22%20wrs%3Abaseline%3D%2240%22%3E%3C!–MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmstyle%20mathsize%3D%2216px%22%3E%3Cmi%20mathvariant%3D%22bold%22%3Er%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfenced%20separators%3D%22%7C%22%3E%3Cmtable%3E%3Cmtr%3E%3Cmtd%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmn%3E5%3C%2Fmn%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3C%2Fmtable%3E%3C%2Fmfenced%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmsub%3E%3Cmi%3E%26%23x3BB%3B%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmfenced%20separators%3D%22%7C%22%3E%3Cmtable%3E%3Cmtr%3E%3Cmtd%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmn%3E4%3C%2Fmn%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3C%2Fmtable%3E%3C%2Fmfenced%3E%3C%2Fmstyle%3E%3C%2Fmath%3E–%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A’math190bc3972c7934354efb2af01e7’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAAERjdnQgDVUNBwAAAWAAAAA6Z2x5ZoPi2VsAAAGcAAABdWhlYWQQC2qxAAADFAAAADZoaGVhCGsXSAAAA0wAAAAkaG10eE2rRkcAAANwAAAAEGxvY2EAHTwYAAADgAAAABRtYXhwBT0FPgAAA5QAAAAgbmFtZaBxlY4AAAO0AAABn3Bvc3QB9wD6AAAFVAAAACBwcmVwa1uragAABXQAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEADAAAAAIAAgAAgAAACsAPSIS%2F%2F8AAAArAD0iEv%2F%2F%2F9b%2Fxd3xAAEAAAAAAAAAAAAAAVQDLACAAQAAVgAqAlgCHgEOASwCLABaAYACgACgANQAgAAAAAAAAAArAFUAgACrANUBAAErAAcAAAACAFUAAAMAA6sAAwAHAAAzESERJSERIVUCq%2F2rAgD%2BAAOr%2FFVVAwAAAQCAAFUC1QKrAAsASQEYsgwBARQTELEAA%2FaxAQT1sAo8sQMF9bAIPLEFBPWwBjyxDQPmALEAABMQsQEG5LEBARMQsAU8sQME5bELBfWwBzyxCQTlMTATIREzESEVIREjESGAAQBVAQD%2FAFX%2FAAGrAQD%2FAFb%2FAAEAAAIAgADrAtUCFQADAAcAZRgBsAgQsAbUsAYQsAXUsAgQsAHUsAEQsADUsAYQsAc8sAUQsAQ8sAEQsAI8sAAQsAM8ALAIELAG1LAGELAH1LAHELAB1LABELAC1LAGELAFPLAHELAEPLABELAAPLACELADPDEwEyE1IR0BITWAAlX9qwJVAcBV1VVVAAEAgAFVAtUBqwADADAYAbAEELEAA%2FawAzyxAgf1sAE8sQUD5gCxAAATELEABuWxAAETELABPLEDBfWwAjwTIRUhgAJV%2FasBq1YAAAAAAQAAAAEAANV4zkFfDzz1AAMEAP%2F%2F%2F%2F%2FWOhNz%2F%2F%2F%2F%2F9Y6E3MAAP8gBIADqwAAAAoAAgABAAAAAAABAAAD6P9qAAAXcAAA%2F7YEgAABAAAAAAAAAAAAAAAAAAAABANSAFUDVgCAA1YAgANWAIAAAAAAAAAAKAAAAKEAAAErAAABdQABAAAABABeAAUAAAAAAAIAgAQAAAAAAAQAAN4AAAAAAAAAFQECAAAAAAAAAAEAEgAAAAAAAAAAAAIADgASAAAAAAAAAAMAMAAgAAAAAAAAAAQAEgBQAAAAAAAAAAUAFgBiAAAAAAAAAAYACQB4AAAAAAAAAAgAHACBAAEAAAAAAAEAEgAAAAEAAAAAAAIADgASAAEAAAAAAAMAMAAgAAEAAAAAAAQAEgBQAAEAAAAAAAUAFgBiAAEAAAAAAAYACQB4AAEAAAAAAAgAHACBAAMAAQQJAAEAEgAAAAMAAQQJAAIADgASAAMAAQQJAAMAMAAgAAMAAQQJAAQAEgBQAAMAAQQJAAUAFgBiAAMAAQQJAAYACQB4AAMAAQQJAAgAHACBAE0AYQB0AGgAIABGAG8AbgB0AFIAZQBnAHUAbABhAHIATQBhAHQAaABzACAARgBvAHIAIABNAG8AcgBlACAATQBhAHQAaAAgAEYAbwBuAHQATQBhAHQAaAAgAEYAbwBuAHQAVgBlAHIAcwBpAG8AbgAgADEALgAwTWF0aF9Gb250AE0AYQB0AGgAcwAgAEYAbwByACAATQBvAHIAZQAAAwAAAAAAAAH0APoAAAAAAAAAAAAAAAAAAAAAAAAAALkHEQAAjYUYALIAAAAVFBOxAAE%2F)format(‘truetype’)%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A’brack_sm2882ad605b1e27be87c7468’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7PH4UAAADMAAAATmNtYXA3kjw6AAABHAAAADxjdnQgAQYDiAAAAVgAAAASZ2x5ZkyYQ7YAAAFsAAAAkWhlYWQLyR8fAAACAAAAADZoaGVhAq0XCAAAAjgAAAAkaG10eDEjA%2FUAAAJcAAAADGxvY2EAAEKZAAACaAAAABBtYXhwBJsEcQAAAngAAAAgbmFtZW7QvZAAAAKYAAAB5XBvc3QArQBVAAAEgAAAACBwcmVwu5WEAAAABKAAAAAHAAACDAGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg9AMD%2FP%2F8AAABVAABAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACI5wjn%2F%2F%2FAAAjnCOf%2F%2F%2FcZdxjAAEAAAAAAAAAAAFUAFQBAAArAIwAgACoAAcAAAACAAAAAADVAQEAAwAHAAAxMxEjFyM1M9XVq4CAAQHWqwABAAAAAABVAVgAAwAfGAGwAy%2BwADyxAgL1sAE8ALEDAD%2BwAjx8sQAG9bABPBEzESNVVQFY%2FqgAAQDXAAABLAFUAAMAIBgBsAUvsAE8sAI8sQAC9bADPACwAy%2BwAjyxAAH1sAE8EzMRI9dVVQFU%2FqwAAAAAAQAAAAEAAIsesexfDzz1AAMEAP%2F%2F%2F%2F%2FVre5k%2F%2F%2F%2F%2F9Wt7mT%2FgP%2F%2FAdYBWAAAAAoAAgABAAAAAAABAAABVP%2F%2FAAAXcP%2BA%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%2F8AAo2FAA%3D%3D)format(‘truetype’)%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A’bracketse552f5417ff4680c6b50499’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7RIisAAADMAAAATmNtYXBi7uzYAAABHAAAAExjdnQgBAkDLgAAAWgAAAASZ2x5Zo64f%2BkAAAF8AAABSWhlYWQLniGcAAACyAAAADZoaGVhBK4XLAAAAwAAAAAkaG10eCWq%2F90AAAMkAAAAFGxvY2EAABknAAADOAAAABhtYXhwBJIESAAAA1AAAAAgbmFtZRAA8I4AAANwAAAB3nBvc3QBwwDgAAAFUAAAACBwcmVwupWEAAAABXAAAAAHAAACggGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg9AMEAAAAAAADgAAAAAAAAAACAAEAAQAAABQAAwABAAAAFAAEADgAAAAKAAgAAgACI5sjnSOeI6D%2F%2FwAAI5sjnSOeI6D%2F%2F9xm3GXcZdxkAAEAAAAAAAAAAAAAAAABVABWAQAALACoA4AAMgAHAAAAAgAAACoA1QNVAAMABwAANTMRIxMjETPV1auAgCoDK%2F0AAtUAAQAAAAABgAOAAAUAJBgBsAAvsAPFsQEC%2FbADELEEBP0AsQAAP7ABPLEEBj%2BwAzwwMTEzEAEjAFUBKyv%2BqwH8AYT%2BqgABAAAAAAGAA4AABQAmGAGwAC6wA8WxBQL9sAMQsQIE%2FQB8sQAGPxiwBTyxAwb2sAI8MDE