equations-of-planes

Equation of a plane in vector form

How do I find the vector equation of a plane?

• A plane is a flat surface which is two-dimensional

  • Imagine a flat piece of paper that continues on forever in both directions

• A plane in often denoted using the capital Greek letter Π

• The vector form of the equation of a plane can be found using two direction vectors on the plane

  • The direction vectors must be

    • parallel to the plane

    • not parallel to each other

    • therefore they will intersect at some point on the plane

• The formula for finding the vector equation of a plane is

  • 

    • Where r is the position vector of any point on the plane

    • a is the position vector of a known point on the plane

    • b and c are two non-parallel direction (displacement) vectors parallel to the plane

    • s and t are scalars

• The formula can also be written as

  • 

    • Where r is the position vector of any point on the plane

    • a, b, c are the position vectors of known points on the plane

    • λ and μ are scalars

  • These formulae are given in the formula booklet but you must make sure you know what each part means

• As a could be the position vector of any point on the plane and b and c could be any non-parallel direction vectors on the plane there are infinite vector equations for a single plane

How do I determine whether a point lies on a plane?

• Given the equation of a plane <img alt=”bold italic r blank equals blank open parentheses fraction numerator bold italic a subscript 1 over denominator table row cell bold italic a subscript 2 end cell row cell bold italic a subscript 3 end cell end table end fraction close parentheses plus lambda open parentheses fraction numerator bold italic b subscript 1 over denominator table row cell bold italic b subscript 2 end cell row cell bold italic b subscript 3 end cell end table end fraction close parentheses plus blank mu open parentheses fraction numerator bold italic c subscript 1 over denominator table row cell bold italic c subscript 2 end cell row cell bold italic c subscript 3 end cell end table end fraction close parentheses” data-mathml='<math ><semantics><mstyle mathsize=”16px”><mi mathvariant=”bold-italic”>r</mi><mi mathvariant=”bold-italic”> </mi><mo>=</mo><mi mathvariant=”bold-italic”> </mi><mfenced separators=”|”><mfrac linethickness=”0pt”><msub><mi mathvariant=”bold-italic”>a</mi><mn>1</mn></msub><mtable><mtr><mtd><msub><mi mathvariant=”bold-italic”>a</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi mathvariant=”bold-italic”>a</mi><mn>3</mn></msub></mtd></mtr></mtable></mfrac></mfenced><mo>+</mo><mi>λ</mi><mfenced separators=”|”><mfrac linethickness=”0pt”><msub><mi mathvariant=”bold-italic”>b</mi><mn>1</mn></msub><mtable><mtr><mtd><msub><mi mathvariant=”bold-italic”>b</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi mathvariant=”bold-italic”>b</mi><mn>3</mn></msub></mtd></mtr></mtable></mfrac></mfenced><mo>+</mo><mi> </mi><mi>μ</mi><mfenced separators=”|”><mfrac linethickness=”0pt”><msub><mi mathvariant=”bold-italic”>c</mi><mn>1</mn></msub><mtable><mtr><mtd><msub><mi mathvariant=”bold-italic”>c</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi mathvariant=”bold-italic”>c</mi><mn>3</mn></msub></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%22222%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%3Cmi%20mathvariant%3D%22bold-italic%22%3Er%3C%2Fmi%3E%3Cmi%20mathvariant%3D%22bold-italic%22%3E%26%23xA0%3B%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmi%20mathvariant%3D%22bold-italic%22%3E%26%23xA0%3B%3C%2Fmi%3E%3Cmfenced%20separators%3D%22%7C%22%3E%3Cmfrac%20linethickness%3D%220pt%22%3E%3Cmsub%3E%3Cmi%20mathvariant%3D%22bold-italic%22%3Ea%3C%2Fmi%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmtable%3E%3Cmtr%3E%3Cmtd%3E%3Cmsub%3E%3Cmi%20mathvariant%3D%22bold-italic%22%3Ea%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsub%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmsub%3E%3Cmi%20mathvariant%3D%22bold-italic%22%3Ea%3C%2Fmi%3E%3Cmn%3E3%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%3Cmsub%3E%3Cmi%20mathvariant%3D%22bold-italic%22%3Eb%3C%2Fmi%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmtable%3E%3Cmtr%3E%3Cmtd%3E%3Cmsub%3E%3Cmi%20mathvariant%3D%22bold-italic%22%3Eb%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsub%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmsub%3E%3Cmi%20mathvariant%3D%22bold-italic%22%3Eb%3C%2Fmi%3E%3Cmn%3E3%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%23xA0%3B%3C%2Fmi%3E%3Cmi%3E%26%23x3BC%3B%3C%2Fmi%3E%3Cmfenced%20separators%3D%22%7C%22%3E%3Cmfrac%20linethickness%3D%220pt%22%3E%3Cmsub%3E%3Cmi%20mathvariant%3D%22bold-italic%22%3Ec%3C%2Fmi%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmtable%3E%3Cmtr%3E%3Cmtd%3E%3Cmsub%3E%3Cmi%20mathvariant%3D%22bold-italic%22%3Ec%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsub%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmsub%3E%3Cmi%20mathvariant%3D%22bold-italic%22%3Ec%3C%2Fmi%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmsub%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%2BAAAACBwcmVwa1uragAABRgAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAA