Page 111 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

2.3 solve problems involving the addition, sub- traction, and scalar multiplication of vectors, including problems arising from real-world applications
Sample problem: A plane on a heading of
N 27° E has an air speed of 375 km/h. The wind is blowing from the south at 62 km/h. Determine the actual direction of travel of the plane and its ground speed.
2.4 perform the operation of dot product on two
vectors represented as directed line segments
→→→→
(i.e., using a•b =|a||b|cosθ) and in
→→
Cartesian form (i.e., using a • b = a1b1 + a2b2 or →→
a • b = a1b1 + a2b2 + a3b3 ) in two-space and three-space, and describe applications of
the dot product (e.g., determining the angle between two vectors; determining the projec-
tion of one vector onto another)
Sample problem: Describe how the dot pro- duct can be used to compare the work done in pulling a wagon over a given distance in a specific direction using a given force for different positions of the handle.
2.5 determine, through investigation, properties of the dot product (e.g., investigate whether it is commutative, distributive, or associative; investigate the dot product of a vector with itself and the dot product of orthogonal vectors)
Sample problem: Investigate geometrically and algebraically the relationship between the dot product of the vectors (1, 0, 1) and
(0, 1, – 1) and the dot product of scalar multi- ples of these vectors. Does this relationship apply to any two vectors? Find a vector that is orthogonal to both the given vectors.
2.6 perform the operation of cross product
on two vectors represented in Cartesian
form in three-space [i.e., using
→→
a x b = (a2b3 – a3b2, a3b1 – a1b3, a1b2 – a2b1)], determine the magnitude of the cross product
→→ →→
(i.e., using|a x b|=|a||b|sinθ), and describe
applications of the cross product (e.g., deter- mining a vector orthogonal to two given vec- tors; determining the turning effect [or torque] when a force is applied to a wrench at differ- ent angles)
Sample problem: Explain how you maximize the torque when you use a wrench and how the inclusion of a ratchet in the design of a wrench helps you to maximize the torque.
2.7 determine, through investigation, properties of the cross product (e.g., investigate whether it is commutative, distributive, or associative; investigate the cross product of collinear vectors)
Sample problem: Investigate algebraically the
relationship between the cross product of
→→ thevectorsa =(1,0,1)andb =(0,1,–1)
and the cross product of scalar multiples
→→
of a and b. Does this relationship apply to any two vectors?
2.8 solve problems involving dot product and cross product (e.g., determining projections, the area of a parallelogram, the volume of a parallelepiped), including problems arising from real-world applications (e.g., determin- ing work, torque, ground speed, velocity, force)
Sample problem: Investigate the dot products →→→ →→→ → a•(axb) and b•(axb) for any two vectors a
→
and b in three-space. What property of the →→
cross product a x b does this verify?
By the end of this course, students will:
3.1 recognize that the solution points (x, y) in two-space of a single linear equation in two variables form a line and that the solution points (x, y) in two-space of a system of two linear equations in two variables determine the point of intersection of two lines, if the lines are not coincident or parallel
Sample problem: Describe algebraically the situations in two-space in which the solution points (x, y) of a system of two linear equa- tions in two variables do not determine a point.
3.2 determine, through investigation with technol- ogy (i.e., 3-D graphing software) and without technology, that the solution points (x, y, z) in three-space of a single linear equation in three variables form a plane and that the solution points (x, y, z) in three-space of a system of two linear equations in three variables form the line of intersection of two planes, if the planes are not coincident or parallel
Sample problem: Use spatial reasoning to compare the shapes of the solutions in three- space with the shapes of the solutions in two- space for each of the linear equations x = 0,
GEOMETRY AND ALGEBRA OF VECTORS
3. Describing Lines and Planes Using Linear Equations
109
Calculus and Vectors
MCV4U