A plane
contains the point
and has a normal vector
a)
Find the equation of the plane in its Cartesian form.
A second point
has coordinates
.
b)
Determine whether point B lies on the same plane.
A plane
has equation ![bold italic r equals open parentheses table row 3 row 3 row 2 end table close parentheses plus lambda space open parentheses table row cell negative 2 end cell row 5 row 3 end table close parentheses plus mu space open parentheses table row 5 row 2 row 7 end table close 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A line with equation
intersects
at a point
.
a)
Write down the equations of the line and the plane in their parametric forms.
b)
Given that the coordinates of
![straight Q](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%2222%22%20width%3D%2214%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%20mathvariant%3D%22normal%22%3EQ%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%226.5%22%20y%3D%2216%22%3EQ%3C%2Ftext%3E%3C%2Fsvg%3E)
are
![open parentheses table row cell 10 comma end cell cell negative 2 comma space 4 end cell end table close parentheses](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%2228%22%20width%3D%2292%22%20wrs%3Abaseline%3D%2220%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfenced%3E%3Cmtable%3E%3Cmtr%3E%3Cmtd%3E%3Cmn%3E10%3C%2Fmn%3E%3Cmo%3E%2C%3C%2Fmo%3E%3C%2Fmtd%3E%3Cmtd%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmn%3E4%3C%2Fmn%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3C%2Fmtable%3E%3C%2Fmfenced%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A'math189201d0dea031dbbb328557e9a'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADxjdnQgDVUNBwAAAVgAAAA6Z2x5ZoPi2VsAAAGUAAAAnGhlYWQQC2qxAAACMAAAADZoaGVhCGsXSAAAAmgAAAAkaG10eE2rRkcAAAKMAAAADGxvY2EAHTwYAAACmAAAABBtYXhwBT0FPgAAAqgAAAAgbmFtZaBxlY4AAALIAAABn3Bvc3QB9wD6AAAEaAAAACBwcmVwa1uragAABIgAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACACwiEv%2F%2FAAAALCIS%2F%2F%2F%2F1d3wAAEAAAAAAAAAAAFUAywAgAEAAFYAKgJYAh4BDgEsAiwAWgGAAoAAoADUAIAAAAAAAAAAKwBVAIAAqwDVAQABKwAHAAAAAgBVAAADAAOrAAMABwAAMxEhESUhESFVAqv9qwIA%2FgADq%2FxVVQMAAAEAVf9kANUAgAAKAAAzNTMVFAYHJz4BN1WALy8bHh4BgHo9URQpDjQxAAEAgAFVAtUBqwADADAYAbAEELEAA%2FawAzyxAgf1sAE8sQUD5gCxAAATELEABuWxAAETELABPLEDBfWwAjwTIRUhgAJV%2FasBq1YAAQAAAAEAANV4zkFfDzz1AAMEAP%2F%2F%2F%2F%2FWOhNz%2F%2F%2F%2F%2F9Y6E3MAAP8gBIADqwAAAAoAAgABAAAAAAABAAAD6P9qAAAXcAAA%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%3D%3D)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A'round_brackets22549f92a457f2409'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMjxkLiAAAADMAAAATmNtYXDf7xCrAAABHAAAADxjdnQgBAkDLgAAAVgAAAASZ2x5Zr5a4R4AAAFsAAABKWhlYWQO9ymoAAACmAAAADZoaGVhDVUVZQAAAtAAAAAkaG10eCFnAAIAAAL0AAAADGxvY2EAAARdAAADAAAAABBtYXhwBIgEWQAAAxAAAAAgbmFtZXHO2TgAAAMwAAACOXBvc3QEagIzAAAFbAAAACBwcmVwupWEAAAABYwAAAAHAAAGrgGQAAUAAAgACAAAAAAACAAIAAAAAAAAAQIAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAo8AMGe%2F57AAAIzgIPAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACACgAKf%2F%2FAAAAKAAp%2F%2F%2F%2F2f%2FZAAEAAAAAAAAAAAFUAFYBAAAsAKgDgAAyAAcAAAACAAAAKgDVA1UAAwAHAAA1MxEjEyMRM9XVq4CAKgMr%2FQAC1QABAAD%2BYAJQCGAACQBNGAGwChCwA9SwAxCwAtSwChCwBdSwBRCwANSwAxCwBzywAhCwCDwAsAoQsAPUsAMQsAfUsAoQsAXUsAoQsADUsAMQsAI8sAcQsAg8MTATAgEzABMQASMABAQBwJD%2BPAQBwJD%2BRANg%2FHD%2BkAFwA5ADoAFg%2FqAAAQAA%2FmACUAhgAAkATRgBsAoQsAPUsAMQsALUsAoQsAXUsAUQsADUsAMQsAc8sAIQsAg8ALAKELAD1LADELAH1LAKELAF1LAKELAA1LADELACPLAHELAIPDEwARIBIwADEAEzAAJMBP5AkAHEBP5AkAG8A2D8cP6QAXADkAOgAWD%2BoAAAAAABAAAAAQAALiwXwl8PPPUAAwgA%2F%2F%2F%2F%2F9Wt7vT%2F%2F%2F%2F%2F1a3u9AAA%2FmAEhAhgAAAACgACAAEAAAAAAAEAAAjO%2FfEAABdwAAD%2F%2FgSEAAEAAAAAAAAAAAAAAAAAAAADANUAAAJQAAACUAAAAAAAAAAAACQAAACmAAABKQABAAAAAwAKAAIAAAAAAAIAgAQAAAAAAAQAAE0AAAAAAAAAFQECAAAAAAAAAAEAPgAAAAAAAAAAAAIADgA%2BAAAAAAAAAAMAXABMAAAAAAAAAAQAPgCoAAAAAAAAAAUAFgDmAAAAAAAAAAYAHwD8AAAAAAAAAAgAHAEbAAEAAAAAAAEAPgAAAAEAAAAAAAIADgA%2BAAEAAAAAAAMAXABMAAEAAAAAAAQAPgCoAAEAAAAAAAUAFgDmAAEAAAAAAAYAHwD8AAEAAAAAAAgAHAEbAAMAAQQJAAEAPgAAAAMAAQQJAAIADgA%2BAAMAAQQJAAMAXABMAAMAAQQJAAQAPgCoAAMAAQQJAAUAFgDmAAMAAQQJAAYAHwD8AAMAAQQJAAgAHAEbAFIAbwB1AG4AZAAgAGIAcgBhAGMAawBlAHQAcwAgAHcAaQB0AGgAIABhAHMAYwBlAG4AdAAgADIAMgA1ADQAUgBlAGcAdQBsAGEAcgBNAGEAdABoAHMAIABGAG8AcgAgAE0AbwByAGUAIABSAG8AdQBuAGQAIABiAHIAYQBjAGsAZQB0AHMAIAB3AGkAdABoACAAYQBzAGMAZQBuAHQAIAAyADIANQA0AFIAbwB1AG4AZAAgAGIAcgBhAGMAawBlAHQAcwAgAHcAaQB0AGgAIABhAHMAYwBlAG4AdAAgADIAMgA1ADQAVgBlAHIAcwBpAG8AbgAgADIALgAwUm91bmRfYnJhY2tldHNfd2l0aF9hc2NlbnRfMjI1NABNAGEAdABoAHMAIABGAG8AcgAgAE0AbwByAGUAAAAAAwAAAAAAAARnAjMAAAAAAAAAAAAAAAAAAAAAAAAAALkH%2FwABjYUA)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22round_brackets22549f92a457f2409%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%223.5%22%20y%3D%2221%22%3E(%3C%2Ftext%3E%3Ctext%20font-family%3D%22round_brackets22549f92a457f2409%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%2287.5%22%20y%3D%2221%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, find the values for
![beta comma space lambda](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%2222%22%20width%3D%2230%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3E%26%23x3B2%3B%3C%2Fmi%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmi%3E%26%23x3BB%3B%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A'math1f7177163c833dff4b38fc8d287'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADRjdnQgDVUNBwAAAVAAAAA6Z2x5ZoPi2VsAAAGMAAAAUmhlYWQQC2qxAAAB4AAAADZoaGVhCGsXSAAAAhgAAAAkaG10eE2rRkcAAAI8AAAACGxvY2EAHTwYAAACRAAAAAxtYXhwBT0FPgAAAlAAAAAgbmFtZaBxlY4AAAJwAAABn3Bvc3QB9wD6AAAEEAAAACBwcmVwa1uragAABDAAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACAAAAAEAAQAAQAAACz%2F%2FwAAACz%2F%2F%2F%2FVAAEAAAAAAAABVAMsAIABAABWACoCWAIeAQ4BLAIsAFoBgAKAAKAA1ACAAAAAAAAAACsAVQCAAKsA1QEAASsABwAAAAIAVQAAAwADqwADAAcAADMRIRElIREhVQKr%2FasCAP4AA6v8VVUDAAABAFX%2FZADVAIAACgAAMzUzFRQGByc%2BATdVgC8vGx4eAYB6PVEUKQ40MQAAAAEAAAABAADVeM5BXw889QADBAD%2F%2F%2F%2F%2F1joTc%2F%2F%2F%2F%2F%2FWOhNzAAD%2FIASAA6sAAAAKAAIAAQAAAAAAAQAAA%2Bj%2FagAAF3AAAP%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%2F)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%224.5%22%20y%3D%2216%22%3E%26%23x3B2%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1f7177163c833dff4b38fc8d287%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2212.5%22%20y%3D%2216%22%3E%2C%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2223.5%22%20y%3D%2216%22%3E%26%23x3BB%3B%3C%2Ftext%3E%3C%2Fsvg%3E)
and
![mu](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%2222%22%20width%3D%2212%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3E%26%23x3BC%3B%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%225.5%22%20y%3D%2216%22%3E%26%23x3BC%3B%3C%2Ftext%3E%3C%2Fsvg%3E)
at the point of intersection.
Consider the two planes
and
which can be defined by the equations
a)
Write down expressions for the normal vectors of each of the two planes.
b)
Hence find the angle between the two planes. Give your answer in radians.
The points
and
have position vectors
and
respectively, relative to the origin
.
The position vectors are given by
a)
Find the direction vectors
![AB with rightwards arrow on top](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%2226%22%20width%3D%2226%22%20wrs%3Abaseline%3D%2220%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmover%3E%3Cmi%3EAB%3C%2Fmi%3E%3Cmo%3E%26%23x2192%3B%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A'horizontal225a4f6397f19c438fc1d'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMlJxIOAAAADMAAAATmNtYXCwg0j%2FAAABHAAAAERjdnQgAAcAAAAAAWAAAAACZ2x5ZuwE1kUAAAFkAAAA3GhlYWQNciGiAAACQAAAADZoaGVhBusXCAAAAngAAAAkaG10eGjtB6sAAAKcAAAAEGxvY2EAAYSYAAACrAAAABRtYXhwBLEEbgAAAsAAAAAgbmFtZReja%2F8AAALgAAAByXBvc3QB9wD6AAAErAAAACBwcmVwu5WEAAAABMwAAAAHAAACRwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAACOv8xEDev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEADAAAAAIAAgAAgAAI6%2FyAPIL%2F%2F8AACOv8gDyC%2F%2F%2F3FIOAg34AAEAAAAAAAAAAAAHAAAAAQAAAVUBqwGrAAMAIxgBsAQQsAE8sAEQsALUsAIQsAU8ALAEELAB1LABELAA1DAxERUhNQGrAatWVgABAAAAqwErAlUACABBGAGwCRCwA9SwAxCwAtSwAhCwBTywCRCwBzwAsAkQsAfUsAcQsAjUsAgQsAHUsAEQsALUsAcQsAXUsAUQsATUMDETJzUNATU3IzVgYAEr%2FtVgYAGqQGvV1WpAVgABAIEBVgGsAawAAwAdGAGwBRCwADywABCwAdQAsAUQsADUsAAQsAPUMTABITUhAaz%2B1QErAVZWAAEAAAABAABFUxdzXw889QADBAD%2F%2F%2F%2F%2F1a3u0f%2F%2F%2F%2F%2FVre7RAAAAAAMAAwAAAAAKAAIAAQAAAAAAAQAAA%2Bj%2FagAAF3AAAP%2F%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%2FwACjYUA)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%222.5%22%20y%3D%228%22%3E%26%23xF20B%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%226.5%22%20y%3D%228%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2210.5%22%20y%3D%228%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2214.5%22%20y%3D%228%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2218.5%22%20y%3D%228%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2222.5%22%20y%3D%228%22%3E%26%23xF200%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%2212.5%22%20y%3D%2220%22%3EAB%3C%2Ftext%3E%3C%2Fsvg%3E)
and
![AC with rightwards arrow on top](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%2226%22%20width%3D%2226%22%20wrs%3Abaseline%3D%2220%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmover%3E%3Cmi%3EAC%3C%2Fmi%3E%3Cmo%3E%26%23x2192%3B%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A'horizontal225a4f6397f19c438fc1d'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMlJxIOAAAADMAAAATmNtYXCwg0j%2FAAABHAAAAERjdnQgAAcAAAAAAWAAAAACZ2x5ZuwE1kUAAAFkAAAA3GhlYWQNciGiAAACQAAAADZoaGVhBusXCAAAAngAAAAkaG10eGjtB6sAAAKcAAAAEGxvY2EAAYSYAAACrAAAABRtYXhwBLEEbgAAAsAAAAAgbmFtZReja%2F8AAALgAAAByXBvc3QB9wD6AAAErAAAACBwcmVwu5WEAAAABMwAAAAHAAACRwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAACOv8xEDev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEADAAAAAIAAgAAgAAI6%2FyAPIL%2F%2F8AACOv8gDyC%2F%2F%2F3FIOAg34AAEAAAAAAAAAAAAHAAAAAQAAAVUBqwGrAAMAIxgBsAQQsAE8sAEQsALUsAIQsAU8ALAEELAB1LABELAA1DAxERUhNQGrAatWVgABAAAAqwErAlUACABBGAGwCRCwA9SwAxCwAtSwAhCwBTywCRCwBzwAsAkQsAfUsAcQsAjUsAgQsAHUsAEQsALUsAcQsAXUsAUQsATUMDETJzUNATU3IzVgYAEr%2FtVgYAGqQGvV1WpAVgABAIEBVgGsAawAAwAdGAGwBRCwADywABCwAdQAsAUQsADUsAAQsAPUMTABITUhAaz%2B1QErAVZWAAEAAAABAABFUxdzXw889QADBAD%2F%2F%2F%2F%2F1a3u0f%2F%2F%2F%2F%2FVre7RAAAAAAMAAwAAAAAKAAIAAQAAAAAAAQAAA%2Bj%2FagAAF3AAAP%2F%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%2FwACjYUA)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%222.5%22%20y%3D%228%22%3E%26%23xF20B%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%226.5%22%20y%3D%228%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2210.5%22%20y%3D%228%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2214.5%22%20y%3D%228%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2218.5%22%20y%3D%228%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2222.5%22%20y%3D%228%22%3E%26%23xF200%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%2212.5%22%20y%3D%2220%22%3EAC%3C%2Ftext%3E%3C%2Fsvg%3E)
.
Points
,
and
all lie on a single plane.
b)
Use the results from part (a) to write down the vector equation of the plane.
c)
Find the Cartesian equation of the plane.
A plane lies parallel to the line with equation
and contains the points
and
with coordinates
and
respectively.
a)
Find the vector
![PX with rightwards arrow on top](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%2226%22%20width%3D%2224%22%20wrs%3Abaseline%3D%2220%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmover%3E%3Cmi%3EPX%3C%2Fmi%3E%3Cmo%3E%26%23x2192%3B%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A'horizontal225a4f6397f19c438fc1d'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMlJxIOAAAADMAAAATmNtYXCwg0j%2FAAABHAAAAERjdnQgAAcAAAAAAWAAAAACZ2x5ZuwE1kUAAAFkAAAA3GhlYWQNciGiAAACQAAAADZoaGVhBusXCAAAAngAAAAkaG10eGjtB6sAAAKcAAAAEGxvY2EAAYSYAAACrAAAABRtYXhwBLEEbgAAAsAAAAAgbmFtZReja%2F8AAALgAAAByXBvc3QB9wD6AAAErAAAACBwcmVwu5WEAAAABMwAAAAHAAACRwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAACOv8xEDev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEADAAAAAIAAgAAgAAI6%2FyAPIL%2F%2F8AACOv8gDyC%2F%2F%2F3FIOAg34AAEAAAAAAAAAAAAHAAAAAQAAAVUBqwGrAAMAIxgBsAQQsAE8sAEQsALUsAIQsAU8ALAEELAB1LABELAA1DAxERUhNQGrAatWVgABAAAAqwErAlUACABBGAGwCRCwA9SwAxCwAtSwAhCwBTywCRCwBzwAsAkQsAfUsAcQsAjUsAgQsAHUsAEQsALUsAcQsAXUsAUQsATUMDETJzUNATU3IzVgYAEr%2FtVgYAGqQGvV1WpAVgABAIEBVgGsAawAAwAdGAGwBRCwADywABCwAdQAsAUQsADUsAAQsAPUMTABITUhAaz%2B1QErAVZWAAEAAAABAABFUxdzXw889QADBAD%2F%2F%2F%2F%2F1a3u0f%2F%2F%2F%2F%2FVre7RAAAAAAMAAwAAAAAKAAIAAQAAAAAAAQAAA%2Bj%2FagAAF3AAAP%2F%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%2FwACjYUA)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%223.5%22%20y%3D%228%22%3E%26%23xF20B%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%227.5%22%20y%3D%228%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2211.5%22%20y%3D%228%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2215.5%22%20y%3D%228%22%3E%26%23x23AF%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22horizontal225a4f6397f19c438fc1d%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2219.5%22%20y%3D%228%22%3E%26%23xF200%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%2211.5%22%20y%3D%2220%22%3EPX%3C%2Ftext%3E%3C%2Fsvg%3E)
.
b)
By appropriate use of the vector product, find the normal to the plane.
c)
Hence find the Cartesian equation of the plane.
Consider the plane defined by the Cartesian equation
a)
Show that the line with equation
![bold italic r space equals space open parentheses table row 3 row 0 row 2 end table close parentheses plus lambda space open parentheses table row 1 row 4 row cell negative 7 end cell end table close 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lies in the plane.
b)
Show that the line with Cartesian equation
![x minus 2 equals fraction numerator y minus 6 over denominator 2 end fraction equals 2 minus z](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%2247%22%20width%3D%22150%22%20wrs%3Abaseline%3D%2230%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmrow%3E%3Cmi%3Ey%3C%2Fmi%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmn%3E6%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmfrac%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmi%3Ez%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A'math135b31cfba37a56451b4768509d'%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADxjdnQgDVUNBwAAAVgAAAA6Z2x5ZoPi2VsAAAGUAAAA%2FGhlYWQQC2qxAAACkAAAADZoaGVhCGsXSAAAAsgAAAAkaG10eE2rRkcAAALsAAAADGxvY2EAHTwYAAAC%2BAAAABBtYXhwBT0FPgAAAwgAAAAgbmFtZaBxlY4AAAMoAAABn3Bvc3QB9wD6AAAEyAAAACBwcmVwa1uragAABOgAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACAD0iEv%2F%2FAAAAPSIS%2F%2F%2F%2FxN3wAAEAAAAAAAAAAAFUAywAgAEAAFYAKgJYAh4BDgEsAiwAWgGAAoAAoADUAIAAAAAAAAAAKwBVAIAAqwDVAQABKwAHAAAAAgBVAAADAAOrAAMABwAAMxEhESUhESFVAqv9qwIA%2FgADq%2FxVVQMAAAIAgADrAtUCFQADAAcAZRgBsAgQsAbUsAYQsAXUsAgQsAHUsAEQsADUsAYQsAc8sAUQsAQ8sAEQsAI8sAAQsAM8ALAIELAG1LAGELAH1LAHELAB1LABELAC1LAGELAFPLAHELAEPLABELAAPLACELADPDEwEyE1IR0BITWAAlX9qwJVAcBV1VVVAAEAgAFVAtUBqwADADAYAbAEELEAA%2FawAzyxAgf1sAE8sQUD5gCxAAATELEABuWxAAETELABPLEDBfWwAjwTIRUhgAJV%2FasBq1YAAQAAAAEAANV4zkFfDzz1AAMEAP%2F%2F%2F%2F%2FWOhNz%2F%2F%2F%2F%2F9Y6E3MAAP8gBIADqwAAAAoAAgABAAAAAAABAAAD6P9qAAAXcAAA%2F7YEgAABAAAAAAAAAAAAAAAAAAAAAwNSAFUDVgCAA1YAgAAAAAAAAAAoAAAAsgAAAPwAAQAAAAMAXgAFAAAAAAACAIAEAAAAAAAEAADeAAAAAAAAABUBAgAAAAAAAAABABIAAAAAAAAAAAACAA4AEgAAAAAAAAADADAAIAAAAAAAAAAEABIAUAAAAAAAAAAFABYAYgAAAAAAAAAGAAkAeAAAAAAAAAAIABwAgQABAAAAAAABABIAAAABAAAAAAACAA4AEgABAAAAAAADADAAIAABAAAAAAAEABIAUAABAAAAAAAFABYAYgABAAAAAAAGAAkAeAABAAAAAAAIABwAgQADAAEECQABABIAAAADAAEECQACAA4AEgADAAEECQADADAAIAADAAEECQAEABIAUAADAAEECQAFABYAYgADAAEECQAGAAkAeAADAAEECQAIABwAgQBNAGEAdABoACAARgBvAG4AdABSAGUAZwB1AGwAYQByAE0AYQB0AGgAcwAgAEYAbwByACAATQBvAHIAZQAgAE0AYQB0AGgAIABGAG8AbgB0AE0AYQB0AGgAIABGAG8AbgB0AFYAZQByAHMAaQBvAG4AIAAxAC4AME1hdGhfRm9udABNAGEAdABoAHMAIABGAG8AcgAgAE0AbwByAGUAAAMAAAAAAAAB9AD6AAAAAAAAAAAAAAAAAAAAAAAAAAC5BxEAAI2FGACyAAAAFRQTsQABPw%3D%3D)format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%224.5%22%20y%3D%2230%22%3Ex%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2218.5%22%20y%3D%2230%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%2231.5%22%20y%3D%2230%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2244.5%22%20y%3D%2230%22%3E%3D%3C%2Ftext%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%2255.5%22%20x2%3D%2294.5%22%20y1%3D%2223.5%22%20y2%3D%2223.5%22%2F%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2261.5%22%20y%3D%2216%22%3Ey%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2275.5%22%20y%3D%2216%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%2288.5%22%20y%3D%2216%22%3E6%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%2275.5%22%20y%3D%2241%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22105.5%22%20y%3D%2230%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%22118.5%22%20y%3D%2230%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22math135b31cfba37a56451b4768509d%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22131.5%22%20y%3D%2230%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22144.5%22%20y%3D%2230%22%3Ez%3C%2Ftext%3E%3C%2Fsvg%3E)
is parallel to the plane but does not lie in the plane.
Consider the planes
and
, which are defined by the equations
a)
By solving the system of equations represented by the three planes show that the system of equations has a unique solution.
b)
Hence write down the coordinates of any point(s) where all three planes intersect.
Consider the line
with vector equation
and the plane
with Cartesian equation
.
a)
Find the angle in radians between the line
![straight L](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%2222%22%20width%3D%2212%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%20mathvariant%3D%22normal%22%3EL%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%225.5%22%20y%3D%2216%22%3EL%3C%2Ftext%3E%3C%2Fsvg%3E)
and the normal to the plane
![capital pi](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%2222%22%20width%3D%2215%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3E%26%23x3A0%3B%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%226.5%22%20y%3D%2216%22%3E%26%23x3A0%3B%3C%2Ftext%3E%3C%2Fsvg%3E)
.
b)
Hence find the angle in radians between the line
![straight L](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%2222%22%20width%3D%2212%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%20mathvariant%3D%22normal%22%3EL%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%225.5%22%20y%3D%2216%22%3EL%3C%2Ftext%3E%3C%2Fsvg%3E)
and the plane
![capital pi](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%2222%22%20width%3D%2215%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3E%26%23x3A0%3B%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%226.5%22%20y%3D%2216%22%3E%26%23x3A0%3B%3C%2Ftext%3E%3C%2Fsvg%3E)
.
Two planes
and
are defined by the equations
a)
Write down expressions for the normal vectors of each of the two planes.
b)
Find the cross product of the two normal vectors.
c)
Find the coordinates of a point that lies on both planes.
d)
Hence find a vector equation of the line of intersection of the two planes.
A line
is defined by the Cartesian equation
and a plane
is defined by the Cartesian equation
, where
is a real constant.
The line
lies in the plane
.
a)
Use the fact that the line
![straight L subscript 1](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%2228%22%20width%3D%2219%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsub%3E%3Cmi%20mathvariant%3D%22normal%22%3EL%3C%2Fmi%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmsub%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%225.5%22%20y%3D%2216%22%3EL%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2214.5%22%20y%3D%2224%22%3E1%3C%2Ftext%3E%3C%2Fsvg%3E)
lies in the plane
![capital pi](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%2222%22%20width%3D%2215%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3E%26%23x3A0%3B%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%226.5%22%20y%3D%2216%22%3E%26%23x3A0%3B%3C%2Ftext%3E%3C%2Fsvg%3E)
to find the value of the constant
![d](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%2222%22%20width%3D%2211%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ed%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%224.5%22%20y%3D%2216%22%3Ed%3C%2Ftext%3E%3C%2Fsvg%3E)
.
Another line,
, passes through the origin and is perpendicular to the plane
.
b)
Write down the equation of line
![straight L subscript 2](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%2228%22%20width%3D%2219%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsub%3E%3Cmi%20mathvariant%3D%22normal%22%3EL%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsub%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%225.5%22%20y%3D%2216%22%3EL%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2214.5%22%20y%3D%2224%22%3E2%3C%2Ftext%3E%3C%2Fsvg%3E)
in vector form.
c)
By considering the parametric form of the equation for
![straight L subscript 2](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%2228%22%20width%3D%2219%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsub%3E%3Cmi%20mathvariant%3D%22normal%22%3EL%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsub%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%225.5%22%20y%3D%2216%22%3EL%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2214.5%22%20y%3D%2224%22%3E2%3C%2Ftext%3E%3C%2Fsvg%3E)
, or otherwise, determine the point of intersection between line
![straight L subscript 2](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%2228%22%20width%3D%2219%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsub%3E%3Cmi%20mathvariant%3D%22normal%22%3EL%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsub%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%225.5%22%20y%3D%2216%22%3EL%3C%2Ftext%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2213%22%20text-anchor%3D%22middle%22%20x%3D%2214.5%22%20y%3D%2224%22%3E2%3C%2Ftext%3E%3C%2Fsvg%3E)
and the plane
![capital pi](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%2222%22%20width%3D%2215%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3E%26%23x3A0%3B%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%226.5%22%20y%3D%2216%22%3E%26%23x3A0%3B%3C%2Ftext%3E%3C%2Fsvg%3E)
.
d)
Hence determine the minimum distance between the plane
![capital pi](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%2222%22%20width%3D%2215%22%20wrs%3Abaseline%3D%2216%22%3E%3C!--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3E%26%23x3A0%3B%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%2F%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%226.5%22%20y%3D%2216%22%3E%26%23x3A0%3B%3C%2Ftext%3E%3C%2Fsvg%3E)
and the origin.