Announcements. Topics: Homework: - section 12.4 (cross/vector product) - section 12.5 (equations of lines and planes)

Transcription

1 Topics: Announcements - section 12.4 (cross/vector product) - section 12.5 (equations of lines and planes) Homework: ü review lecture notes thoroughl ü work on eercises from the tetbook in sections 12.4 and 12.5 ü work on Assignment 12 (all questions ecept #11)

2 Equations of Lines in 2D Recall: To write an equation of a line in the -plane, we need to know a point ( 1, 1 ) on the line and the slope, m, of the line. oint-slope equation: 1 = m( 1 ) oint--intercept equation: = m + b Eample: =

3 Vector Equation of a Line in 2D If 0 ( 0, 0 ) is a known point on the line and (,) is an arbitrar point on the line, then b the triangle law for vector addition, we can write that O O O = O O

4 r 0 Vector Equation of a Line in 2D Let vector v=<a,b> be an vector parallel to the line (often called a direction vector for the line). 0 tv If we let r be the the position vector O and r 0 be the position vector O 0, then the vector equation of the line can be written as O r v = a,b r = r 0 + tv where t is called a parameter and can be an real number.

5 arametric Equations of a Line in 2D If we epress the vectors algebraicall, we have that, = 0, 0 + t a,b 0 t a,b 0, 0, Separating the and components, we obtain the parametric equations of the line: O v = a,b = 0 + at = 0 + bt

6 Smmetric Equation of a Line in 2D If we then solve these equations for t (provided that the direction numbers, a and b, are not zero) and equate the components, we obtain the smmetric equation for the line: O 0, 0 0 t a,b, v = a,b 0 a = 0 b

7 Equations of Lines in 2D Eample 1: Find vector and parametric equations of the line =

8 Equations of Lines in 2D Eample 2: Find vector and parametric equations of the line through points (-4,3) and Q(2,-1).

9 Equations of Lines in 2D Eample 3: (a) Find a vector equation of the line through the point (1,5) and perpendicular to the line r = 1,2 + t 4,3. (b) Write the smmetric equation for this line.

10 Vector Equation of a Line in 3D If 0 ( 0, 0, z 0 ) is a known point on the line and (,,z) is an arbitrar point on the line, then b the triangle law for vector addition, we can write that 0 O 0 z 0 O O O = O 0 + 0

11 Vector Equation of a Line in 3D Let vector v=<a,b,c> be an vector parallel to the line (often called a direction vector for the line). 0 z tv If we let r be the the position vector O and r 0 be the position vector O 0, then the vector equation of the line can be written as r 0 O r v = a,b,c r = r 0 + tv where t is called a parameter and can be an real number.

12 arametric Equations of a Line in 3D If we epress the vectors algebraicall, we have that,, z = 0, 0, z 0 + t a, b, c 0 z t a,b,c,,z Separating the,, and z components, we obtain the parametric equations of the line: 0, 0,z 0 O v = a,b,c = 0 + at = 0 + bt z = z 0 + ct

13 Smmetric Equations of a Line in 3D If we then solve these equations for t (provided that the direction numbers, a, b, and c are not zero) and equate the components, we obtain the smmetric equations for the line: 0 0, 0,z 0 z t a,b,c O,,z v = a,b,c 0 a = 0 b = z z 0 c

14 Equations of Lines in 3D Eample 1: Find vector, parametric, and smmetric equations of the line through (1,0,3) and parallel to the line r = 1, 6,0 + t 5,7, 1.

15 Equations of Lines in 3D Eample 2: Find vector, parametric, and smmetric equations of the line through points (1,2,0) and Q(-3,2,1).

16 Equation of a lane A plane is determined b a point 0 ( 0, 0, z 0 ) in the plane and a vector n, called a normal vector, that is perpendicular to the plane.

17 Equation of a lane Let n=<a,b,c> be a normal vector for the plane. This normal vector, or simpl normal, is perpendicular to ever vector in the plane. If 0 ( 0, 0, z 0 ) is a known point in the plane and (,,z) is an arbitrar point in the plane, then n is perpendicular to 0 and so we have that n 0 = 0

18 Equation of a lane Using the definition of the dot product for algebraic vectors, we obtain the scalar equation of the plane through 0 ( 0, 0, z 0 ) with normal vector n=<a,b,c>: a( 0 ) + b( 0 ) + c( 0 ) = 0 This equation is often epanded and simplified to a + b + cz + d = 0 where d = (a 0 + b 0 + cz 0 ).

19 Eample #26. Find an equation of the plane through the point (2,0,1) and perpendicular to the line =3t, =2-t, z=3+4t. Equations of lanes

20 Eample #32. Find an equation of the plane through the origin and the points (3,-2,1) and (1,1,1). Equations of lanes

21 Equations of lanes Eample #36. Find an equation of the plane that passes through the point (6,-1,3) and contains the line /3=+4=z/2.