Date Lesson TOPIC Homework. Parametric and Vector Equations of a Line in R 2 Pg. 433 # 2 6, 9, 11. Vector and Parametric Equation of a Plane in Space

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1 UNIT 3 - EQUATIONS OF LINES AND PLANES Date Lessn TOPIC Hmewrk Sept. 29 Oct.3 Oct.4 Oct (19) 3.2 (20) 3.3 (21) 3.4 (22) OPT Parametric and Vectr Equatins f a Line in R 2 Pg. 433 # 2 6, 9, 11 Scalar r Cartesian Equatin f a Line Pg. 443 # 1, 2, 3, 5, 6, 8, 9 (10ab, 11b) see Ex. 6 n pg. 441 Equatins f a Line in R 3 Pg. 449 # 4, 5, 6, 9, 12 Vectr and Parametric Equatin f a Plane in Space Pg. 459 # 2, 6, 7, 9, 10, 13, 15 Mid-Chapter Review Pg. 451 # 1-22 Oct (23) 8.5 The Scalar r Cartesian Equatin f a Plane Ax By Cz D 0 Pg. 468 # 5, 7, 9b, 11, 13, 15 Oct.11 Oct (24) 3.7 (25) Review fr Unit 3 Test Pg. 480 # 3, 4, 5, 7, 8, 10, 13, 16, 20c, 21a, 23, 31b, 34abf TEST- UNIT 3

2 MCV 4U Lessn 3.1 Parametric and Vectr Equatins f a Line Slpe-Intercept Vectr Parametric Equatins f Lines in Tw-Space y mx b r r td, t R r ( x, y) ( x, y ) t( a, b), t R x x y y at bt m is the slpe f the line b is the y-intercept r (x, y) is a psitin vectr t any pint n the line r (x, y )is a psitin vectr t a knwn pint n the line d (a, b) is a directin vectr fr the line. State d in lwest terms, n fractins. t R t is the parameter, Scalar/Cartesian By C 0 Symmetric x x a Ax n ( A,B)is a nrmal vectr t y y b t, a, b 0 the line Rearrange each parametric equatin fr t. If ne cmpnent f the directin vectr is 0, n symmetric equatin exists. A nrmal vectr t a line is perpendicular t that line. T define a line in tw space, yu need: tw pints n the line, r a pint and a directin vectr, r a perpendicular vectr and a pint. NOTE: There is n distinct vectr r parametric equatin fr any line. s r r d The psitin vectr r has its tip n the line. The directin vectr d is parallel t the line. The vectr s is a scalar multiple f the directin vectr d. r = r + s and s is a scalar multiple f d, we can say that s = t d. r = r + t d.

3 Cincident vs. Parallel Lines Bth same directin vectr Cincident Lines have cmmn pints Parallel Lines have n cmmn pints Ex. 1 A line passes thrugh pints A (1, 4) and B (3, 1), a) Write a vectr equatin fr the line. b) Determine three mre psitin vectrs fr the line. c) Determine if the pint (2, 3) is n the line. When we separated the vectr equatin int tw parts, ne fr each variable, we used the parametric equatins f the line. They are called parametric because the result is gverned by the parameter t, t R

4 Ex. 2 Cnsider the line l 1. l 1: x 3 2t y 5 4t l 2: x 1 3t y 8 12t a) Find the crdinates f tw pints n the line. b) Write a vectr equatin f the line. c) Determine if line l 1 is parallel t l 2. Ex. 3 A line passes thrugh the pints A( 2, 3) and B(5, 2). a) Write a set f parametric equatins fr the line. b) Determine if C(26, 1) and D(36, 3) are n the line.

5 Ex. 4. Given r (0,3) t(1,5), determine if it is cincident t the line with x symmetric equatin 2 y Pg. 433 # 2 6, 9, 11

6 MCV 4U Lessn 3.2 Scalar r Cartesian Equatin f a Line in R 2 Scalar/Cartesian Equatin Ax By C 0 n ( A,B)is a nrmal vectr t the line Cnverting frm ne frm f Equatin t anther. Scalar t vectr: Get nrmal n ( A,B) Get directin d (B, A) Get any pint n the line. - ne with a zer crdinate is usually easy Use integer values if pssible Get vectr equatincnvert t parametric r symmetric Ex. Cnvert 2x + 5y 9 = 0 t parametric frm. n (2, 5) d ( 5, 2) (x, y) = (2, 1) r (2, 1) + ( 5, 2)t x = 2 5t y = 1 + 2t Vectr/parametric/symmetric t Scalar: write in symmetric frm - even if a r b is 0 Crss multiply and simplify Ex. 1 Find the scalar equatin f the line with nrmal (2,3) which passes thrugh P 1 (4, 1).

7 Ex. 2 a) Determine the vectr equatin f a line that is nrmal t 4x + 3y 12 = 0 if it passes thrugh P(7, 1). b) Determine the parametric equatins f the line. Pg. 443 # 1, 2, 3, 5, 6, 8, 9 (10ab, 11b) see Ex. 6 n pg. 441

8 MCV 4U Lessn 3.3 Equatins f a Line in Three Space Equatins f a Line in Three Space r r td r (x, y, z) is a psitin vectr t any pint n the line. Vectr r Parametric (x, y, z) (x, y, z ) t (a,b,c) x x at y y bt z z ct r (x, y, z ) is a psitin vectr t a knwn pint n the line. Symmetric x x a y y b z z c d (a,b,c) is a directin vectr fr the line. If 2 f a, b, r c = 0, then There is n symmetric equatin. There is NO scalar/cartesian equatin f a line in three-space. Ex. 1 Write the vectr, parametric, and symmetric equatins f the line passing thrugh P( 1, 0, 5) and with directin vectr ( 1, 4, 1).

9 Ex. 2 Determine if the fllwing pairs f lines are cincident, parallel, r neither. a) r 1 (1,1,1) s(6,2,0) r 2 (5, 3,1) t (9, 3,0) b) x y 4 z r 4 (3, 5,2) t(2, 4,1) Ex. 3 Sketch the line r (3,2,5) t(1,9,6). Plt the pint (3, -2, 5). Find anther pint by subbing in a value fr t, t 0. Draw line thrugh the 2 pints. Pg. 449 # 4, 5, 6, 9, 12

10 MCV 4U Lessn 3.4 The Vectr and Parametric Equatins f a Plane in Space In tw space, a scalar equatin defines a line. In three space, a scalar equatin defines a plane. In three space, a plane can be defined by a vectr equatin, parametric equatins, r a scalar equatin. Equatins f Planes in Three Space r r sa tb Vectr r (x, y, z) (x, y, z ) s(a 1, a 2, a 3 ) t(b 1,b 2,b 3 ) r (x, y, z) is a psitin vectr fr any pint n the plane. r (x, y, z ) is a psitin vectr fr a knwn pint n the plane. Parametric x x sa 1 tb 1 y y sa 2 tb 2 z z sa 3 tb 3 a (a 1, a 2, a 3 ) and b (b 1, b 2,b 3 ) are nn parallel directins vectrs parallel t the plane. s and t are scalars, s,t R (A, B, C ) is a vectr nrmal t the plane Scalar/Cartesian Ax By Cz D 0 A plane can be uniquely defined by three nn-cllinear pints r by a pint and tw nn parallel directin vectrs Fr vectr and parametric equatins, any cmbinatin f values f the parameters s and t will pduce a pint n the plane. Fr a scalar equatin, any pint (x, y, z) that satisfies the equatin lies n the plane. The x intercept f a plane is fund by setting y = z = 0 and slving fr x. Similarly, the y and z intercepts are fund by setting x =z = 0 and x = y = 0, respectively.

11 Ex. 1 Cnsider the plane with directin vectrs a (8, 5, 4) and b (1, 3, 2) thrugh P (3, 7, 0). a) Write the vectr and parametric equatins f the plane. b) Determine if the pint Q ( 10, 8, 6) is n the plane. c) Find the crdinates f anther pint n the plane. d) Find the x intercept f the plane.

12 Ex. 2 Find the vectr equatin f each plane. a) the plane with x int = A(2, 0, 0), y int = B(0, 4, 0) and z int = C(0, 0, 5). b) the plane cntaining the line (x, y, z) = (0, 3, 5) + t(6, 2, 1) and parallel t the line (x, y, z) = (1, 7, 4) + s(1, 3, 3). Ex. 3 Find the equatin f the plane that cntains the tw parallel line r 1 (3,1,5) t(1,2,3) r 2 (0,2,4) t(3,6,9) Pg. 459 # 2, 6, 7, 9, 10, 13, 15

13 MCV 4U Lessn 3.5 Caretesian (Scalar) Equatins f a Plane in Space If P(x, y, z) is any pint n a plane, P x, y, z ) is a specific pint n the plane, and n ( A, B, C) is the ( nrmal t the plane, then PP n 0 ( x x, y y, z z ) ( A, B, C) 0 Ax Ax By By Cz Cz 0 Ax By Cz Ax By Cz 0, Ax By Cz R Ax By Cz D 0 ***NOTE: There is nly ne unique scalar equatin fr each plane. All ther equatins are simply scalar multiples. T Generate a Scalar Equatin frm Parametric r Vectr Equatins Find the crss prduct f the directin vectrs t find the nrmal. Write the scalar equatin with the nrmal (A, B, C). Substitute any pint n the plane t find D. Write the equatin.

14 Ex. 1 Find the scalar equatin f the plane r ( 4,2,1) s(0,1,2) t( 2,1,1 ). Ex. 2 Cnvert 4x + 5y - 2z + 8 = 0 t parametric frm. Pg. 468 # 5, 7, 9b, 11, 13, 15

Transitive property: If arb and brc, then arc.

Transitive property: If arb and brc, then arc. Principles f Gemetry Chapter 1 t 4 Summary Sheet Definitins Chapter 1.2 Cllinear pints are pints that lie n the same line. Chapter 1.3 An issceles triangle is a triangle that has tw cngruent sides. A line