STRAIGHT LINES & PLANES PARAMETRIC EQUATIONS OF LINES The lie "L" is parallel to the directio vector "v". A fixed poit: "( a, b, c) " o the lie is give. Positio vectors are draw from the origi to the fixed poit ad a arbitrary poit: " ( x, y, z) " o the lie. The differece betwee these two vectors lies o the lie ad is parallel to "v". r r r0 r0 tv + tv x a + t v x y b + t v y z c + t v z Page of 4
Example # : Fid parametric equatios of the lie that passes thru the give poit ad is parallel to the give vector. ( a, b, c) (, 4, 3) v i j xt () + t yt () 4 t zt () 3 + t Lie Geerated by Parametric Equatios + k z, 4, 3 y x Page of 4
Example # : Show that the give lies itersect ad fid the poit of itersectio. L : x + t y + 3 t z 3 + t L : x + t y 3 + 4 t z 4 + t Lies Itersectig i 3-Space z,, x y The lies itersect where the coordiates are the same. So, set the correspodig coordiates equal to each other. Page 3 of 4
+ t + t + 3 t 3 + 4 t 3 + t 4 + t Solve these equatios ad fid this result. t t That result yields this for the poit of itersectio: (,, ). Example # 3: Show that the give lies are parallel. L : x t y t z + t L : x + t y 3 4 t z 5 t Page 4 of 4
Parallel Lies i 3-Space z y x The directio vectors for the lies are the same. 4 Therefore, the lies are parallel. Page 5 of 4
Equatios of Plaes Positio vectors are draw from the origi to the fixed poit: " ( a, b, c) " ad a arbitrary poit: " ( x, y, z) " o the plae. The differece betwee these two vectors lies i the plae. A vector, " ", the plae ormal, is perpedicular to the plae. Page 6 of 4
Sice the differece vector lies i the plae ad the ormal vector is perpedicular to the plae, the dot product of the two is zero. r r0 0 r x y r 0 z a b r c r0 x y z a b c r r0 α x a α β γ + β y b + γ z c 0 α x a + β y b + γ z c 0 Page 7 of 4
Example # 4: Fid a equatio of the plae that passes thru the three give poits. Q3 (,, 6) R5 (,, 0) P3,, The swiftest approach is to fid a plae ormal by takig the cross product of two vectors formed from the three poits. PR 4 PQ 4 4 PR PQ 4 4 4 0 4 α β γ α x a + β y b + γ z c 0 x a 0 y b 4 z c 0 Choose ayoe of the three poits to substitute for the fixed poit (a, b, c ). x + 0 y 3 + 4 z 0 Page 8 of 4
3 Poits Determie Plae Q z P R y x Example # 5: Fid a equatio of the plae that cotais the give lie ad is perpedicular to the give plae. Lie: x 3 t y + t z + t Plae: x + y z Here is a ormal vector for the give plae. g Page 9 of 4
A ormal vector for the ukow plae ca be foud by takig the cross product of the ormal vector for the give plae with the directio vector of the lie. 3 α x a + β y b + γ z c 0 3 5 4 3 x + 5 y + 4 z + 0 Lie & Plae Determie Plae g x L z y Page 0 of 4
Distaces i 3-Space Example # 6: Fid the distace betwee the poit "P" ad the lie thru the poits "A" & "B'. A (,, 0) B(, 3, 4) P 3,, Here are the parametric equatios of the lie ad the poit. L: x t P: x t () 3t yt () 3 + t zt () 3 yt () zt BA 3 PA 4 0 4 () () 4 4t BA PA BA PA si( θ) d PA si θ BA BA PA Page of 4
3 4 d 4.4 3 4 4 0 Poit-to-Lie z P 3,, y x Page of 4
Example # 7: Fid the distace betwee the poit "P" ad the plae. x P,, 3 y + z 4 Let the poit: Q lie i the plae. Qx 0, y 0, z 0 That meas that "Q" satisfies the equatio of the plae. x 0 y 0 + z 0 4 Here is a ormal vector of the plae. Form this vector. QP x 0 y 0 3 z 0 Page 3 of 4
Proj QP Proj QP ( QP ) ( ) x 0 y 0 3 z 0 Proj QP 9 x 0 + y 0 z 0 9 9 9 But, x 0 y 0 + z 0 4 Page 4 of 4
Therefore, x 0 y 0 + z 0 4 Proj QP 9 4 9 9 9 0 9 0 9 5 9 d Proj QP 0 9 0 9 5 9.667 d.667 Page 5 of 4
Poit-to-Plae z y x Page 6 of 4
Example # 8: Fid the distace betwee the two give parallel lies. L: x() t t yt () t zt () + t L: x() t + t yt () 3 4 t zt () 5 t Pick a poit that lies o oe of the two lies. "P" lies o L. P0,, Now, fid the distace betwee L ad "P" just as you did i Example # 6. Here are two poits that lie o L. B3 (,, 3) A, 3, 5 BA 4 PA 3 4 Page 7 of 4
BA PA BA PA si( θ) d PA si θ BA BA PA d 4 4.45 4 3 Page 8 of 4
Lie-to-Lie: Parallel Lies P0,, z y x Page 9 of 4
Example # 9: Fid the distace betwee the two give skew lies. L: x() t + 7 t yt () 3 + t zt () 5 3 t L: x() t 4 t yt () 6 zt () 7 + t v 7 v 3 0 The cross-product of the directio vectors for the lies is perpedicular to them ad defies a ormal to the plae formed by the two lies ad a ifiity of plaes parallel to that plae. v v 7 3 0 Page 0 of 4
This is the equatio for all of these plaes. x x 0 y y 0 + z z 0 0 Now pick a poit o either of the lies. For example, choose the poit (, 3, 5 ) that lies o L. The this is the oe plae amog all those with the same ormal that also cotais L. x y 3 + z 5 0 Now choose ay poit o L, say ( 4, 6, 7 ), ad fid the distace betwee that poit ad the plae just as you did i Example # 7. x P46,, 7 y + z 6 Let the poit: Q lie i the plae. Qx 0, y 0, z 0 That meas that "Q" satisfies the equatio of the plae. x 0 y 0 + z 0 6 Page of 4
Here is a ormal vector of the plae. Form this vector. QP 4 x 0 6 y 0 7 z 0 Proj QP Proj QP ( QP ) ( ) 4 x 0 6 y 0 7 z 0 Page of 4
Proj QP 5 x 0 + y 0 z 0 6 6 6 But, x 0 y 0 + z 0 6 Therefore, x 0 + y 0 z 0 6 Proj QP 5 + 6 6 6 6 50 6 75 6 6 Page 3 of 4
d Proj QP 50 6 75.8 6 6 Lie-toLie: Skew Lies P46,, 7 z y x Page 4 of 4