Page 110 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Grade 12, University Preparation
1. demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications;
2. perform operations on vectors in two-space and three-space, and use the properties of these operations to solve problems, including those arising from real-world applications;
3. distinguish between the geometric representations of a single linear equation or a system of two linear equations in two-space and three-space, and determine different geometric configurations of lines and planes in three-space;
4. represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections.
1. Representing Vectors Geometrically and Algebraically
2. Operating With Vectors
C. GEOMETRY AND ALGEBRA OF VECTORS
OVERALL EXPECTATIONS
By the end of this course, students will:
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
SPECIFIC EXPECTATIONS
By the end of this course, students will:
1.1 recognize a vector as a quantity with both magnitude and direction, and identify, gather, and interpret information about real-world applications of vectors (e.g., displacement, forces involved in structural design, simple animation of computer graphics, velocity determined using GPS)
Sample problem: Position is represented using vectors. Explain why knowing that someone is 69 km from Lindsay, Ontario, is not sufficient to identify their exact position.
1.2 represent a vector in two-space geometrically as a directed line segment, with directions ex- pressed in different ways (e.g., 320o; N 40o W),
and algebraically (e.g., using Cartesian coordi- nates; using polar coordinates), and recognize vectors with the same magnitude and direc- tion but different positions as equal vectors
1.3 determine, using trigonometric relationships [e.g.,x=rcosθ,y=rsinθ,θ=tan–1 (y)or
–1 (y ) √ 2 2 x
tan x + 180o, r = x + y ],
the Cartesian representation of a vector in two-space given as a directed line segment, or
the representation as a directed line segment of a vector in two-space given in Cartesian form [e.g., representing the vector (8, 6) as a directed line segment]
Sample problem: Represent the vector with a magnitude of 8 and a direction of 30o anti- clockwise to the positive x-axis in Cartesian form.
1.4 recognize that points and vectors in three-space can both be represented using Cartesian coor- dinates, and determine the distance between two points and the magnitude of a vector using their Cartesian representations
By the end of this course, students will:
2.1 perform the operations of addition, subtrac- tion, and scalar multiplication on vectors represented as directed line segments in two- space, and on vectors represented in Cartesian form in two-space and three-space
2.2 determine, through investigation with and withouttechnology,someproperties(e.g., commutative, associative, and distributive properties) of the operations of addition, subtraction, and scalar multiplication of vectors
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