MAT 1236 Calculus III Section 12.5 Part I Equations of Line and Planes

HW… WebAssign 12.5 Part I (9 problems 75 min.)

Preview Equations of Lines Vector Equations Parametric Equations Symmetric Equations Equations of Planes

Recall: Position Vectors Given any point, is the position vector of P. To serve as a position vector, the initial point O of the vector is fixed.

Equations of Lines In 2D, what kind of info is required to determine a line? Type 1: Type 2: Q: How to extend these ideas?

Vector Equations

Parametric Equations

Example 1

Example 1: Parametric Equation

Remarks

Example 1: Symmetric Equation

What if… If one of the component is a constant, then…

3 Possible Scenarios Given 2 distinct lines in 3D, they are either

Example 2 Show that the 2 lines are parallel.

Example 2: Plan Show that the 2 lines are parallel.

Expectations Give precise reasons.

Incorrect Logic... Show that the 2 lines are parallel.

Example 3 Find the intersection point of the 2 lines

Example 3

Example 3: Plan

Example 3 Find the intersection point of the 2 lines (Note that it is given that the two lines intersect each other. If this is not given, then the solutions steps will be different.)

Expectations You are expected to carefully explain your solutions. Answers alone are not sufficient for quizzes or exams.

Example 4 Show that the two lines are skew.

Example 4: Plan Show that the two lines are skew. (a) Show that the two lines are not parallel. (b) Show that the two lines have no intersection points.

Example 4: Plan for (b) Show that the two lines are skew.

Expectations