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

HW… WebAssign 12.5 Part II 28 problems, start it ASAP Be sure to think about the solutions method carefully. There can be a lot of variations It is not practical to memorize all the cases. You need to understand the principles and practice solving problems. Hints for the last problem is in the HO.

Quiz Quiz: 12.3, 12.4, 12.5 I

Preview Equations of Planes Vector Equations Scalar Equations Similar development as in lines

Vector Equations of Planes Ingredients A (fixed) point on the plane A (fixed) vector n= orthogonal to the plane n is called a ________________ For a (general) point on the plane, ________________

Scalar Equations

Example 1 Find an equation of the plane through the point P(-5,1,2) with normal vector n=.

Example 1 Find an equation of the plane through the point P(-5,1,2) with normal vector n=. Can you recover n= from the linear equation?

Example 2 Find the equation of the plane through the points A(-1,1,-1), B(1,-1,2), C(4,0,3)

Parallel Planes Two planes are parallel if their normal vectors are_______.

Parallel Planes If two planes are not parallel, then they intersect in a straight line and the angle between the two planes is defined as the acute angle between their normal vectors.

Example 3 (a) Find the angle between the two planes

Example 3 (b) Find symmetric equations of the line of intersection.

Example 3 (b) Method I Find symmetric equations of the line of intersection.

Geometric Meanings Unless the line is horizontal, there is a point on it with a zero z coordinate.

Geometric Meanings Unless the line is horizontal, there is a point on it with a zero z coordinate.

Geometric Meanings Unless the line is horizontal, there is a point on it with a zero z coordinate.

Example 3 (b) Method II

Example 4 Find the distance from a point to the plane (Reading Assignment: read this from the text)