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

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Presentation transcript:

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

HW… WebAssign 12.5 Part I

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 Ingredients A (fixed) point on the line A (fixed) vector v= parallel to the line Any vector parallel to the line can be represented by ________________ The position vector of a (general) point on the line can be represented by ________________

Parametric Equations

Example 1 Find a vector equation and parametric equations for the line that passes through the point (1,1,5) and is parallel to the vector.

Example 1

Example 1: Parametric Equation Can you recover (1,1,5) and from the parametric equation?

Remarks As usual, parametric equations are not unique (e.g. v 1 = gives another parametric equation.)

Example 1: Symmetric Equation

Can you recover (1,1,5) and from the symmetric equation?

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

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

Example 2 Show that the 2 lines are parallel.

Example 3 Find the intersection point of the 2 lines (The lines intersect if there is a pair of parameters (s,t) that gives the same point on the two lines.)

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 Show that the two lines are skew. 1. Show that the two lines are not parallel. 2. Show that the two have no intersection points.

Expectations To show that two lines are non-parallel, you are expected to show that the cross product of the two (direction) vectors is a non-zero vector. Do not substitute s and t directly into the 3 rd equation. You are expected to compute the values of the two sides separately and compare the values.