Page 112 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Grade 12, University Preparation
4. Describing Lines and Planes Using Scalar, Vector, and Parametric Equations
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
y = 0, and y = x. For each of the equations
z = 5, y – z = 3, and x + z = 1, describe the shape of the solution points (x, y, z) in three- space. Verify the shapes of the solutions in three-space using technology.
3.3 determine, through investigation using a variety of tools and strategies (e.g., modelling with cardboard sheets and drinking straws; sketching on isometric graph paper), different geometric configurations of combinations of up to three lines and/or planes in three-space (e.g., two skew lines, three parallel planes, two intersecting planes, an intersecting line and plane); organize the configurations based on whether they intersect and, if so, how they intersect (i.e., in a point, in a line, in a plane)
By the end of this course, students will:
4.1 recognize a scalar equation for a line in two-space to be an equation of the form Ax + By + C = 0, represent a line in
two-space using a vector equation (i.e.,
→→→
r = r0 + tm) and parametric equations, and make connections between a scalar equation, a vector equation, and parametric equations
of a line in two-space
4.2 recognize that a line in three-space cannot
be represented by a scalar equation, and rep- resent a line in three-space using the scalar equations of two intersecting planes and using vector and parametric equations (e.g., given a direction vector and a point on the line, or given two points on the line)
Sample problem: Represent the line passing through (3, 2, – 1) and (0, 2, 1) with the scalar equations of two intersecting planes, with
a vector equation, and with parametric equations.
4.3 recognize a normal to a plane geometrically
(i.e., as a vector perpendicular to the plane) and algebraically [e.g., one normal to the plane 3x + 5y – 2z = 6 is (3, 5, –2)], and deter- mine, through investigation, some geometric properties of the plane (e.g., the direction of any normal to a plane is constant; all scalar multiples of a normal to a plane are also nor- mals to that plane; three non-collinear points determine a plane; the resultant, or sum, of any two vectors in a plane also lies in the plane)
Sample problem: How does the relationship →→→
a • (b x c) = 0 help you determine whether
three non-parallel planes intersect in a point, if
→→→
a, b, and c represent normals to the three planes?
4.4 recognize a scalar equation for a plane in three-space to be an equation of the form
Ax + By + Cz + D = 0 whose solution points make up the plane, determine the intersection of three planes represented using scalar equations by solving a system of three linear equations in three unknowns algebraically (e.g., by using elimination or substitution), and make connections between the algebraic solution and the geometric configuration of the three planes
Sample problem: Determine the equation of a plane P3 that intersects the planes
P1, x + y + z = 1, and P2, x – y + z = 0, in a single point. Determine the equation of a plane P4 that intersects P1 and P2 in more than one point.
4.5 determine, using properties of a plane, the scalar, vector, and parametric equations of a plane
Sample problem: Determine the scalar, vector, and parametric equations of the plane that passes through the points (3, 2, 5), (0, – 2, 2), and (1, 3, 1).
4.6 determine the equation of a plane in its scalar, vector, or parametric form, given another of these forms
Sample problem: Represent the plane
→
r = (2, 1, 0) + s(1, –1, 3) + t(2, 0, –5), where
s and t are real numbers, with a scalar equation.
4.7 solve problems relating to lines and planes in three-space that are represented in a variety of ways (e.g., scalar, vector, parametric equa- tions) and involving distances (e.g., between a point and a plane; between two skew lines) or intersections (e.g., of two lines, of a line and a plane), and interpret the result geometrically
Sample problem: Determine the intersection
of the perpendicular line drawn from the
point A(–5, 3, 7) to the plane
→
v = (0, 0, 2) + t(– 1, 1, 3) + s(2, 0, – 3),
and determine the distance from point A to the plane.
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