Lesson10.notebook November 28, 2012 Unit 2: Lines and Planes in 3 Space Linear Combinations of Vectors Today's goal: I can write vectors as linear combinations of each other using the appropriate method and can use this knowledge to help explain the difference between coplanar and collinear. Thus far we have produced resultants and scalar multiples of vectors. Now we combine the processes. Linear Combination of Vectors involve creating a vector w such that it is created from vectors u and v. w = We say that vectors u, v, and w are coplanar, or all 3 vectors are in the same plane (can be drawn on the same sheet of paper). Vectors are collinear if and only if it is possible to find a non zero scalar such that u = av. Example: Given 5u + v 7w = 0, express each vector as a linear combination of the other 2.
Lesson10.notebook November 28, 2012 Example: For the vectors shown in the parallelogram determine: A) a as a linear combination of b and c. B) a as a linear combination of c and d. a d c C) b as a linear combination of a, c, and d. b Example: For u, v, and w answer the following: Given u = 6, v = 9, w = 12 A) Write w as a linear combination of u and v. B) Write u as a linear combination of v and w. v 30 60 u w
Lesson10.notebook November 28, 2012 Example: For vectors u, v, and w as shown, determine an expression for w as a linear combination of u and v given that: u = 1, v = 5, and w = 8. 120 u v 20 40 w Example: If vector w = [15, 22, 1.5], u = [6, 2, 9], and v = [ 4, 7, 1], write w in terms of u and v. Homework page 207 # 1 11
Lesson11.notebook November 28, 2012 Linear Dependence and Independence Today's goal: I can explain the difference between linear dependence and independence and successfully apply those skills to appropriate problems. Linear Dependence of Two Vectors Two vectors, u and v, are linearly dependent if and only if there are scalars a and b, both non zero, such that au + bv = 0. Two vectors that DO NOT satisfy the above are independent (ie. not collinear). Diagrams: Linear Dependence of Three Vectors Three vectors: u, v, and w, are linearly dependent if and only if there are scalars: a, b, and c, not all zero, such that au + bv + cw = 0. Three vectors who DO NOT satisfy the above are independent. Diagrams:
Lesson11.notebook November 28, 2012 Example: Assume vectors u, v, and w are lineraly independent. Determine the values for the variables. A) (2a + 4)u + (b 3)v = 0. B) (a 7)u + (2a 3)v + 5w = 0 Example: If u = [2, 5], v = [ 3, 1], and w = [4, 2] and are linearly dependent, determine the values of a, b, and c such that au + bv + cw = 0.
Lesson11.notebook November 28, 2012 Example: Show that u = 2i + 5j and v = 3i 2j are linearly independent. Homework: page 213 # 2 5, 7, 10 12
Lesson12.notebook November 28, 2012 Parametric and Vector Equations Today's goal: I understand how vectors can be written in parametric and vector form and I can demonstrate my skill by taking provided information an creating appropriate equations. To determine the equation of a line you need to know: 1) 2) In Grade 9 we know the slope defines the "direction" a line travels. In Grade 12 we now change from slope to direction vector. Example: Find the direction vector for each line. A) A line passing through A(4, 5) and B(3, 7). B) The line l 2 with slope 4 / 5. Example: Develop the General form of the vector equation for the line through the point P o ( 3, 5) with direction vector d = (1, 2). Hint #1: The vector P 0 P is a scalar multiple of the direction vector. Hint #2: What we need to do is to write the vector P 0 P as a linear combination using the origin. Hint #3: The resulting equation is what we call the "General Form" of the vector equation.
Lesson12.notebook November 28, 2012 From the General Form of a vector equation we can generate the parametric equation: And finally from the Parametric Form of a vector equation we can get the Symmetric Form...
Lesson12.notebook November 28, 2012 Example: Determine the General, Parametric, and Symmetric Equation of a vector equation of a line passing through A(1,7) and B(5, 3). Example: Determine a direction vector that is: 1) Parallel to (x, y) = (3, 1) + t(5, 1) 2)Perpendicular to x = 3 4t y = 5 + 3t
Lesson12.notebook November 28, 2012 Example: Determine the vector equation for the following: 1) x 7 = y 2 5 3 2) x = 3 4t y = 2t Homework: page 245 #2 8, 11 16
Lesson13.notebook November 28, 2012 Scalar Equation of a Line Today's goal: I can explain how the Standard Equation was developed and I can demonstrate my understanding by taking given information and developing a vector equation in standard form. In grade 9, the equation Ax + By + C = 0 was called the "Standard Equation" of a line. In grade 12 it is known as the Scalar equation or the Cartesian equation. Remember: The Dot Product is also known as the Scalar Product. Yesterday we talked about the "Direction" vector. In the homework you had to find a vector perpendicular to that vector. The perpendicular vector also has its own name, the "Normal" vector. Developing the Scalar Equation Develop a scalar equation of the line passing through the point P 0 (4, 1) with a normal vector, n = [3, 5].
Lesson13.notebook November 28, 2012 Summary: The Scalar equation can be found using either: P 0 P n = 0 or n 1 x + n 2 + C = 0, where n = [n 1, n 2 ] Example: Find the Scalar equation of the line passing through the point (6, 2) and normal to n = [2, 5].
Lesson13.notebook November 28, 2012 Example: Find a normal vector for the following lines: A) 3x + 2y = 7 B) [x, y] = [2, 5] + t[ 3, 1] Example: For the line 2x + 5y = 16, determine the equation of the line in the General, Parametric, and Symmetric vector form.
Lesson13.notebook November 28, 2012 Distance from a Point to a Line The distance from the point Q to the line Ax + By + C = 0 is determined by: P 0 n Q Example: Determine the distance between the point Q(8, 3) and the line 2x + 5y 16 = 0. Homework: page 251 # 2 9, 11, 12
Lesson14.notebook November 28, 2012 Matrices Today's goal: I can use a matrix to solve a problem requiring the solution of 3 equations with 3 unknowns so that I can use this skill to help interpret and better understand the interactions between lines and planes. Matrices are used to help analyze systems of equations. For systems of equations there are 3 possible outcomes: 1) Independent (A unique solution) 2) Dependent (Infinite number of solutions) 3) Inconsistent (No solution) A matrix is an array of data. Example: Determine the following values from the matrix below.
Lesson14.notebook November 28, 2012 Rules for Matrices and Systems of Equations 1) You can multiply any row by a non zero scalar. 2) You can add / subtract any multiple of one equation to another. 3) You can interchange rows at any time. There are two forms for a solved matrix: To get to these forms we will apply the rules stated above. Here we go! Possible outcomes for 3 x 3 matrices: (interpreted)
Lesson14.notebook November 28, 2012 Example: Solve 3x 4y = 18 2x + 5y = 4 Example: Solve 3x + 8y 3z = 6 2x 3y + z = 0 x + 2y z = 3
Lesson14.notebook November 28, 2012 Example: Solve. 2x 2y + z = 0 x + 3y + 2z = 3 2x + 6y + 12z = 10 Example: Solve. 2x y + 2z = 3 3x + 2y z = 2 5x + 8y 7z = 0 Homework Handout
Lesson15.notebook November 28, 2012 Equations of Lines in 3 Space Today's goal: I can extend my knowledge of vector, parametric and symmetric form of vector equations from 2 space to 3 space and develop appropriate equations from given information. How does 3 Space differ from 2 Space? Example: A line passes through the point P 0 ( 3, 5, 2) and has a direction vector of d = [1, 2, 1]. Determine the equation of the line in Vector, Parametric and Symmetric form.
Lesson15.notebook November 28, 2012 Example: Determine the Vector Equation of a line passing through the points A(1, 3, 2) and B(9, 2, 0). Example: Develop a Symmetric equation that is parallel to the line: x = 3t 5 y = 2 + t and passing through the point P(3, 5, 1). z = t 5
Lesson15.notebook November 28, 2012 Example: Determine the vector equation of a line that passes through the point P(5, 2, 8) and is perpendicular to the lines [x, y] = [4, 3, 2] + t[2, 5, 1] and [x, y] = [ 1, 5, 0] + s[3, 2, 3]. Homework: page 256 # 1 9, 12
Lesson16.notebook December 06, 2012 Intersection of 2 Lines Today's goal: I can use previous skills to determine the solution(s) to a system with 2 lines and describe my answer using words or a picture. There are 4 possible outcomes for lines in three space: Example: Determine the point of intersection (if possible) for the following systems of equations. 1) [x, y] = [1, 7] + t[3, 7] and x 4 = y 1 2
Lesson16.notebook December 06, 2012 2) x + 5 = y 2 = z + 7 and x = y + 6 = z + 3 3 2 6 5 1 3) [x, y, z] = [3, 3, 4] + t[5, 4, 2] and [x, y, z] = [ 2, 1, 0] + s[1, 3, 7] Homework page 263 # 2, 4 8, 12, 13
Lesson17.notebook December 06, 2012 What do we know up to this point...? Vector Equation of a Plane Today's goal: I am able to take given information a create a vector equation of a plane. In order to determine the equation of a plane, the following information is required: 1) 2) Equations of a Plane and Notation
Lesson17.notebook December 06, 2012 Example: Determine the Vector and Parametric equations of a plane passing through the points A(1, 7, 2), B(4, 0, 1) and C(1, 2, 3). Example: Determine two direction vectors for each plane listed below: A) A plane parallel to [x, y, z] = [1, 3, 6] + s[ 3, 4, 0] + t[2, 1, 3] B) The plane passing through P(2, 1, 0) and contains the line x = y 1 = z 1 2 3 C) The xz plane.
Lesson17.notebook December 06, 2012 Example: Determine 2 points on the plane [x, y, z] = [2, 0, 1] + s[0, 2, 1] + t[5, 3, 4] Example: Determine the vector equation of a line passing through the point A(2, 5, 1) and perpendicular to the plane π = [5, 2, 1] + s[ 1, 1 6] + t[3, 2, 5]. Homework page 279 #3 10
Lesson18.notebook December 06, 2012 Scalar Equation of a Plane Today's goal: I can take given information and create a scalar equation of a plane using previous knowledge. Why did we never get introduced to the scalar equation of a line in 3 space? Recall in 2 space: Based on what we did with 2 space we can assume the process is the same: P 0 P n = 0 and the resulting form should be Ax + By + Cz + D = 0 Summary:
Lesson18.notebook December 06, 2012 Example: Develop a scalar equation of the plane through the point P(4, 1, 3) with normal, n = [3, 5, 2]. Example: Write the equation [x, y, z] = [2, 5, 1] + t[1, 1, 1] in scalar form.
Lesson18.notebook December 06, 2012 Example: Find the distance between the point A(3, 1, 1) to the plane 4x 8y z + 41 = 0. Homework: page 286 #2, 5 10, 12, 15
Lesson19.notebook December 06, 2012 Intersection of a Line and a Plane Today's goal: I can determine the solution(s) to the interaction between a line and a plane and describe my results through words or pictures. The 3 possible cases for a line and a plane are: Shortcuts for finding intersection points:
Lesson19.notebook December 06, 2012 Example: Find the intersection between the following lines and planes: 1) [x, y, z] = [3, 2, 1] + t[4, 3, 2] and x y 2z = 0. 2) x 2 = y + 5 = z 6 and 5x + y 2z = 0. 3 8
Lesson19.notebook December 06, 2012 3) x = 2 + t y = 1 2t and [x, y, z] = [2, 1, 1] + m[1, 2, 2] + p[1, 0, 3] z = 3 5t Homework: page 292 #1 3, 5, 8 11
Lesson20.notebook December 06, 2012 Intersection of 2 Planes Today's goal: I can apply the process to determine the solution(s) to 2 planes and interpret my results using words or pictures. The 3 possible cases for 2 planes are: Example: Determine the intersection between the two planes for the following: 1) 4x 5y 2z 1 = 0 and x y 2z = 0 Note: Discuss the relationship between the 2 normals.
Lesson20.notebook December 06, 2012 2) x = 4 + s x = 8 + 6m + 2p π = y = s + 3t π = y = 3m + 5p z = s 2t z = m p 3) x + 2y + 3z 6 = 0 and 4x + 8y + 12z 25 = 0 Homework: page 300 #3 6, 9
Lesson21.notebook December 06, 2012 Intersection of 3 Planes Today's goal: I can use previous skills to determine the solution(s) for 3 planes and use words and diagrams to explain my answer. There are 8 possible cases for 3 planes:
Lesson21.notebook December 06, 2012 Example: Determine the intersection of the following: 1) 3x 3y 2z = 14 5x + y 6z = 10 x 2y + 4z = 9
Lesson21.notebook December 06, 2012 2) 6x + 2y 8z = 18 3x + y 4z = 20 12x + 4y 16z = 18 Homework: page 308 #2 5, 8, 9