AQA Further Pure 4. Vectors. Section 4: Intersections and angles. Finding the intersection of a line and a plane

Transcription

1 AQA Further Pure 4 Vectors Sectio 4: Itersectios ad agles Notes ad Examples These otes cotai subsectios o Fidig the itersectio of a lie ad a plae The itersectio of two plaes The agle betwee a lie ad a plae The agle betwee two plaes The distace of a poit from a plae Fidig the itersectio of a lie ad a plae You ca fid the poit of itersectio of a lie ad a plae by substitutig the vector equatio of the lie ito the vector equatio of the plae, ad solvig to fid the value of (uless the lie is parallel to the plae, i which case there is o value of, or the lie lies i the plae, i which case there are a ifiite umber of possible values of ) You ca the substitute this value of ito the equatio of the lie to fid the coordiates of the poit of itersectio The followig example shows you how to fid the itersectio of a lie ad a plae Example Fid the poit of itersectio of the lie x 3y z 6 r 3 λ ad the plae 4 3 x The geeral poit o the lie is: r y 3 λ z 4 3 So readig across: x λ y 3 λ z 43λ Substitute these ito the equatio of the plae x 3y z 6 : ( λ) 3( 3 λ) (4 3 λ) 6 Simplifyig: MEI, 8/05/09 /5

2 4 λ 9 6λ 4 3λ 6 7 λ 6 λ λ Now substitute λ ito the equatio of the lie to fid the positio vector of the poit of itersectio x r y 3 z 4 3 So the coordiates of the poit of itersectio are: (, -, ) Check this poit lies o the plae x 3y z 6 : 3 ( ) 3 6 as required The itersectio of two plaes The lie of itersectio of two plaes ca be foud i several differet ways Oe of the most efficiet ways uses the vector product The lie of itersectio of two plaes obviously lies i both plaes, so it is perpedicular to the ormals of both plaes Therefore, fidig the vector product of the two ormals gives the directio vector for the lie of itersectio Oce you kow this, you ca use ay poit i both plaes to fid the equatio of the lie Example Fid the vector equatio of the lie of itersectio of the plaes: x y z ad 3x y z 3 The ormal vectors to the plae are ad 3 3 The directio vector of the lie of itersectio 5 7 Choosig x = to fid a poit o both plaes gives yz y z Solvig simultaeously gives y = 0 ad z = - 3 The equatio of the lie is r MEI, 8/05/09 /5

3 You ca check that this equatio is correct by usig it to give the coordiates of ay two poits o the lie ad showig that these poits each lie o both plaes eg gives the poit 4, 5, 8 ad 0 gives the poit, 0, Check for yourself that these poits satisfy both plae equatios Why do you eed two poits to be sure your equatio is correct? The agle betwee a lie ad a plae The agle betwee a lie ad a plae is the complemet of the agle betwee the lie ad the ormal to the plae (ie the agle betwee a lie ad a plae = 90 - the agle betwee the lie ad the ormal to the plae) You ca fid the cosie of the agle betwee the ormal ad the lie by usig the scalar product: For the lie r a b ad the plae r d, the acute agle betwee the directio vector of the lie, b, ad the ormal vector to the plae,, is give by cos b b ad so the acute agle betwee the lie ad the plae is give by 90 - Example 3 Fid the agle betwee the lie r 3 ad the plae 3x y z The directio vector of the lie is b The ormal vector to the plae is 3 b MEI, 8/05/09 3/5

4 b 3 ( 4) 6 3 ( ) ( ) 4 The agle betwee the lie ad the ormal vector is give by cos The agle betwee the lie ad the plae is = 30 The agle betwee two plaes The acute agle betwee two plaes is equal to the acute agle betwee the ormals to the plaes This meas that the agle betwee two plaes ca be foud quite easily usig the scalar product cos Example 4 Fid the acute agle betwee the plaes 3x y 4z ad x y 3z 3 The ormal vectors to the plae are ad ( 4) 6 ( ) 3 cos The acute agle betwee the plaes is The distace of a poit from a plae Fidig shortest distaces is ot specifically required by the syllabus, but you could be led through a questio like the example below The shortest distace of a poit A to a plae is the distace AP where AP is a lie perpedicular to the plae ad P is a poit o the plae MEI, 8/05/09 4/5

5 To do this: Step : Fid the equatio of the lie through A perpedicular to the plae Step : Fid the poit of itersectio, P, of the lie ad the plae Step 3: Fid the distace AP Example 5 Fid the distace of the poit A(-,, 4) from the plae 3x y z 3 Step : The directio vector perpedicular to the plae is: 3 So the equatio of the lie through A(-,, 4) is r λ 4 x 3 Step : The geeral poit o the lie is r y λ z 4 So readig across x 3λ y λ z 4 λ Substitute these ito the equatio of the plae 3x y z : 3( 3 λ) ( λ) (4 λ) Simplifyig: 3 9λ 4 4λ 4 λ 3 4λ 4λ 4 λ Now substitute λ ito the equatio of the lie to fid the positio vector of the poit of itersectio x 3 r y 0 z 4 5 So the coordiates of the poit of itersectio are: P(, 0, 5) Step 3: 3 AP AP 3 ( ) So the distace of the poit A to the plae is 4 uits MEI, 8/05/09 5/5