MCV 4UI THE DISTNCE FOM OINT TO LINE IN - SCE LENING GOL: Be able to erive the forula for the istace fro a poit to a lie i. Be able to calculate the istace fro a poit to a lie i a the istace betwee parallel lies i. Let x y be the equatio of a lie i -space. Call it l.. Let ( ) be ay poit i -space. x, y. ( x, y l ). The the shortest istace,, fro to the lie is the perpeicular istace fro to a poit o the lie, call it. 4. Let ( ) be ay poit o the lie. x, y ( x, y ) l ( x, y ) 5. The istace fro to the lie ca be fou by fiig the agitue of the projectio of oto the oral of the lie,. agitue of the projectio of oto. ( ) x, y l ( x, y )
SO proj oto ( x x, y y ) B) B) x x y By Sice ( x ) is o the lie, x y, y C x By The istace fro the poit ( x ) to the lie x y is:, y x y 6. THE DISTNCE BETWEEN LLEL LINES The istace betwee parallel lies is the sae as the istace fro a poit to a lie. To fi the istace betwee parallel lies, select a poit fro oe of the lies a fi the istace fro that poit to the other lie. 7. Fi the istace fro the poit (4, 5) to the lie x + y 5 x y (4) + ( 5) 5 + 5 The istace is uits. 5
MCV 4UI THE DISTNCE FOM OINT TO LNE IN -SCE LENING GOL: Be able to erive the forula for the istace fro a poit to a plae i. Be able to calculate the istace fro a poit to a plae i a the istace betwee parallel plaes i.. Let x y z + D be the equatio of a plae i -space.. Let be ay poit i -space.,. The shortest istace,, fro a poit to the plae is the perpeicular istace fro to a poit o the plae, call it., 4. Let be ay poit o the plae,, 5. The istace fro to the plae ca be fou by fiig the agitue of the projectio of oto the oral of the plae,. (Just like a pt to a lie above.),
SO p roj oto ( x x, y y, z z ) B, C) B, C) x x y By z Cz Sice is o the plae, x y z + D, D x By Cz The istace fro the poit to the plae x y z + D is:, x y z + D 6. THE DISTNCE BETWEEN LLEL LNES The istace betwee parallel lies is the sae as the istace fro a poit to a lie. To fi the istace betwee parallel lies, select a poit fro oe of the lies a fi the istace fro that poit to the other lie. 7. Fi the istace betwee the parallel plaes, x + y 7z 5 a x + 6y z +. Solutio: poit o the plae, x + y 7z 5 is (5,,). Use this as ( x, y, z ) So you ust use the, B, C, D values fro x + 6y z +.. x y z + D ( ) (5) + 6() + ( ) + 6 + 48 The istace is 486 48 486 uits.
MCV THE DISTNCE FOM OINT TO LINE IN -SCE LENING GOL: Be able to calculate the istace fro a poit to a lie i a the istace betwee parallel lies i. The istace fro the poit to the lie ( x, y, z) ( x, y, z ) + k(,, ) is (, ) is:,,, k, where is the kow poit, a, 6. THE DISTNCE BETWEEN LLEL LINES IN -SCE The istace betwee parallel lies i -space is the sae as the istace fro a poit to a lie i -space. To fi the istace betwee parallel lies, select a poit fro oe of the lies a fi the istace fro that poit to the other lie. 7. Fi the istace fro the poit (,,8) to the lie ( x, y, z) (,7, ) + k(,,9 ) (,,8) (,7, ) (, 9,) (, 9,) (,,9 ) (,,9 ) - -9 - -9 9 (,,7 ) (,,9 ) 58 86 The istace is 58 uits. 86