Unit 3: Lines and Planes

In this unit students will:

• 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

• determine through investigation 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

• determine different geometric configurations of combinations of up to three lines and/or planes in three-space; organize the configurations based on whether they intersect, and, if so, how they intersect

• 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 and parametric equations, and make connections between a scalar equation, a vector equation, and parametric equations of a line in two-space

• recognize that a line in three space cannot be represented by a scalar equation, and represent a line in three space using the scalar equations of two intersecting planes and using vector and parametric equations

• recognize a normal to a plane geometrically and algebraically, and determine some geometric properties of the plane

• recognize a scalar equation fora plane in three space to be an equation of the form Ax + By + Cz + D = 0 whose solutions 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, and make connections between the algebraic solutions and the geometric configuration of three planes

• determine, using properties of a plane, the scalar, vector, or parametric form, given another form

• determine the equation of a plane in its scalar, vector, or parametric form, given another of these forms

• solve problems relation to lines and planes in three space that are represented in a variety of ways and involving distances or intersections, and interpre the results geometrically