combinations-of-lines-and-planes

Intersections of lines & planes

How do I tell if a line is parallel to a plane?

• A line is parallel to a plane if its direction vector is perpendicular to the plane’s normal vector

• If you know the Cartesian equation of the plane in the form  then the values of a, b, and c are the individual components of a normal vector to the plane

• The scalar product can be used to check in the direction vector and the normal vector are perpendicular

  • If two vectors are perpendicular their scalar product will be zero

How do I tell if the line lies in the plane?

• If the line is parallel to the plane then it will either never intersect or it will lie in the plane

  • Check to see if they have a common point

• If a line is parallel to a plane and they share any point, then the line lies in the plane

How do I find the point of intersection of a line and a plane in Cartesian form?

• If a line is not parallel to a plane it will intersect it at a single point

• If both the vector equation of the line and the Cartesian equation of the plane is known then this can be found by:

• STEP 1: Set the position vector of the point you are looking for to have the individual components x, y, and z and substitute into the vector equation of the line

  • <img alt=”begin mathsize 16px style open parentheses fraction numerator x over denominator table row y row z end table end fraction close parentheses equals blank open parentheses fraction numerator x subscript 0 over denominator table row cell y subscript 0 end cell row cell z subscript 0 end cell end table end fraction close parentheses plus lambda open parentheses fraction numerator l over denominator table row m row n end table end fraction close parentheses end style” data-mathml='<math ><semantics><mstyle mathsize=”16px”><mfenced separators=”|”><mfrac linethickness=”0pt”><mi>x</mi><mtable><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mfrac></mfenced><mo>=</mo><mi> </mi><mfenced separators=”|”><mfrac linethickness=”0pt”><msub><mi>x</mi><mn>0</mn></msub><mtable><mtr><mtd><msub><mi>y</mi><mn>0</mn></msub></mtd></mtr><mtr><mtd><msub><mi>z</mi><mn>0</mn></msub></mtd></mtr></mtable></mfrac></mfenced><mo>+</mo><mi>λ</mi><mfenced separators=”|”><mfrac linethickness=”0pt”><mi>l</mi><mtable><mtr><mtd><mi>m</mi></mtd></mtr><mtr><mtd><mi>n</mi></mtd></mtr></mtable></mfrac></mfenced></mstyle><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″}</annotation></semantics></math>’ height=”89″ 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%2289%22%20width%3D%22172%22%20wrs%3Abaseline%3D%2233%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%3Cmfenced%20separators%3D%22%7C%22%3E%3Cmfrac%20linethickness%3D%220pt%22%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmtable%3E%3Cmtr%3E%3Cmtd%3E%3Cmi%3Ey%3C%2Fmi%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmi%3Ez%3C%2Fmi%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3C%2Fmtable%3E%3C%2Fmfrac%3E%3C%2Fmfenced%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmi%3E%26%23xA0%3B%3C%2Fmi%3E%3Cmfenced%20separators%3D%22%7C%22%3E%3Cmfrac%20linethickness%3D%220pt%22%3E%3Cmsub%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmtable%3E%3Cmtr%3E%3Cmtd%3E%3Cmsub%3E%3Cmi%3Ey%3C%2Fmi%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmsub%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmsub%3E%3Cmi%3Ez%3C%2Fmi%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmsub%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3C%2Fmtable%3E%3C%2Fmfrac%3E%3C%2Fmfenced%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3E%26%23x3BB%3B%3C%2Fmi%3E%3Cmfenced%20separators%3D%22%7C%22%3E%3Cmfrac%20linethickness%3D%220pt%22%3E%3Cmi%3El%3C%2Fmi%3E%3Cmtable%3E%3Cmtr%3E%3Cmtd%3E%3Cmi%3Em%3C%2Fmi%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmi%3En%3C%2Fmi%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3C%2Fmtable%3E%3C%2Fmfrac%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’math1564b4c0e54101ac57a0cb68c16’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADxjdnQgDVUNBwAAAVgAAAA6Z2x5ZoPi2VsAAAGUAAABK2hlYWQQC2qxAAACwAAAADZoaGVhCGsXSAAAAvgAAAAkaG10eE2rRkcAAAMcAAAADGxvY2EAHTwYAAADKAAAABBtYXhwBT0FPgAAAzgAAAAgbmFtZaBxlY4AAANYAAABn3Bvc3QB9wD6AAAE%2BAAAACBwcmVwa1uragAABRgAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACACsAPf%2F%2FAAAAKwA9%2F%2F%2F%2F1v%2FFAAEAAAAAAAAAAAFUAywAgAEAAFYAKgJYAh4BDgEsAiwAWgGAAoAAoADUAIAAAAAAAAAAKwBVAIAAqwDVAQABKwAHAAAAAgBVAAADAAOrAAMABwAAMxEhESUhESFVAqv9qwIA%2FgADq%2FxVVQMAAAEAgABVAtUCqwALAEkBGLIMAQEUExCxAAP2sQEE9bAKPLEDBfWwCDyxBQT1sAY8sQ0D5gCxAAATELEBBuSxAQETELAFPLEDBOWxCwX1sAc8sQkE5TEwEyERMxEhFSERIxEhgAEAVQEA%2FwBV%2FwABqwEA%2FwBW%2FwABAAACAIAA6wLVAhUAAwAHAGUYAbAIELAG1LAGELAF1LAIELAB1LABELAA1LAGELAHPLAFELAEPLABELACPLAAELADPACwCBCwBtSwBhCwB9SwBxCwAdSwARCwAtSwBhCwBTywBxCwBDywARCwADywAhCwAzwxMBMhNSEdASE1gAJV%2FasCVQHAVdVVVQAAAQAAAAEAANV4zkFfDzz1AAMEAP%2F%2F%2F%2F%2FWOhNz%2F%2F%2F%2F%2F9Y6E3MAAP8gBIADqwAAAAoAAgABAAAAAAABAAAD6P9qAAAXcAAA%2F7YEgAABAAAAAAAAAAAAAAAAAAAAAwNSAFUDVgCAA1YAgAAAAAAAAAAoAAAAoQAAASsAAQAAAAMAXgAFAAAAAAACAIAEAAAAAAAEAADeAAAAAAAAABUBAgAAAAAAAAABABIAAAAAAAAAAAACAA4AEgAAAAAAAAADADAAIAAAAAAAAAAEABIAUAAAAAAAAAAFABYAYgAAAAAAAAAGAAkAeAAAAAAAAAAIABwAgQABAAAAAAABABIAAAABAAAAAAACAA4AEgABAAAAAAADADAAIAABAAAAAAAEABIAUAABAAAAAAAFABYAYgABAAAAAAAGAAkAeAABAAAAAAAIABwAgQADAAEECQABABIAAAADAAEECQACAA4AEgADAAEECQADADAAIAADAAEECQAEABIAUAADAAEECQAFABYAYgADAAEECQAGAAkAeAADAAEECQAIABwAgQBNAGEAdABoACAARgBvAG4AdABSAGUAZwB1AGwAYQByAE0AYQB0AGgAcwAgAEYAbwByACAATQBvAHIAZQAgAE0AYQB0AGgAIABGAG8AbgB0AE0AYQB0AGgAIABGAG8AbgB0AFYAZQByAHMAaQBvAG4AIAAxAC4AME1hdGhfRm9udABNAGEAdABoAHMAIABGAG8AcgAgAE0AbwByAGUAAAMAAAAAAAAB9AD6AAAAAAAAAAAAAAAAAAAAAAAAAAC5BxEAAI2FGACyAAAAFRQTsQABPw%3D%3D)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%2FQB8sQAGPxiwBTyxAwb2sAI8MDEREgEzABMBAVQr%2FtMCA4D91v6qAYQB%2FAAB%2F6wAAAEsA4AABQAnGAGwAC%2BwBzywA8WxAQL9sAMQsQQE%2FQCxAAA%2FsAE8sQQGP7ADPDAxISMSATMAASxVAf7UKwFVAfwBhP6rAAH%2FrAAAASwDgAAFACkYAbABL7AHPLAExbEAAv2wBBCxAwT9AHyxAQY%2FsAA8GLEEAD%2BwAzwwMRMzEAEjANdV%2FqsrASsDgP3V%2FqsBhgAAAAABAAAAAQAAeuTcpl8PPPUAAwQA%2F%2F%2F%2F%2F9Wt7o7%2F%2F%2F%2F%2F1a3ujv%2BsAAABgAOAAAAACgACAAEAAAAAAAEAAAOAAAAAABdw%2F6z%2FrAGAAAEAAAAAAAAAAAAAAAAAAAAFANUAAAEsAAABLAAAASz%2FrAEs%2F6wAAAAAAAAAJAAAAGgAAACzAAAA%2FQAAAUkAAQAAAAUACAACAAAAAAACAIAEAAAAAAAEAAA%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%2FAAGNhQA%3D)format(‘truetype’)%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22bracketse552f5417ff4680c6b50499%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%221.5%22%20y%3D%2224%22%3E%26%23x239B%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%221.5%22%20y%3D%2229%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%221.5%22%20y%3D%2234%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%221.5%22%20y%3D%2239%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%221.5%22%20y%3D%2244%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%221.5%22%20y%3D%2249%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%221.5%22%20y%3D%2254%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%221.5%22%20y%3D%2259%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22bracketse552f5417ff4680c6b50499%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%221.5%22%20y%3D%2273%22%3E%26%23x239D%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22bracketse552f5417ff4680c6b50499%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2233.5%22%20y%3D%2224%22%3E%26%23x239E%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2233.5%22%20y%3D%2229%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2233.5%22%20y%3D%2234%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2233.5%22%20y%3D%2239%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2233.5%22%20y%3D%2244%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2233.5%22%20y%3D%2249%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2233.5%22%20y%3D%2254%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2233.5%22%20y%3D%2259%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22bracketse552f5417ff4680c6b50499%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2233.5%22%20y%3D%2273%22%3E%26%23x23A0%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2219.5%22%20y%3D%2222%22%3Ex%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2219.5%22%20y%3D%2246%22%3Ey%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2218.5%22%20y%3D%2269%22%3Ez%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1564b4c0e54101ac57a0cb68c16%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%2247.5%22%20y%3D%2233%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22bracketse552f5417ff4680c6b50499%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2214%22%3E%26%23x239B%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2219%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2224%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2229%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2234%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2239%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2244%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2249%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2254%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2259%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2264%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2269%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2274%22%3E%26%23x239C%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22bracketse552f5417ff4680c6b50499%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2261.5%22%20y%3D%2288%22%3E%26%23x239D%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22bracketse552f5417ff4680c6b50499%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2214%22%3E%26%23x239E%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2219%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2224%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2229%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2234%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2239%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2244%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2249%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2254%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2259%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2264%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2269%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2274%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22bracketse552f5417ff4680c6b50499%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%2299.5%22%20y%3D%2288%22%3E%26%23x23A0%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2279.5%22%20y%3D%2215%22%3Ex%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2287.5%22%20y%3D%2222%22%3E0%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2279.5%22%20y%3D%2246%22%3Ey%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2212%22%20text-anchor%3D%22middle%22%20x%3D%2287.5%22%20y%3D%2253%22%3E0%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2278.5%22%