13. 5 Lines and Planes in Space

Assigned questions: 9,11,29,31,33,35,37,43,44,49, 63,69,73,87

Lines in Space

• To find the equation of a line, you need both a point on the line and a vector in the direction of that line

• $\theta = \arccos \frac{a*b}{|a| |b|}$ for the angle between two vectors

Distance from a Point to a Line

Equations of Planes

Plane in $\R^3$: given a fixed point $P_0$ and a nonzero normal vector n, the set of points P in $\R^3$ for which $\vec{P_0P}$ is orthogxonal to n is called a plane

General equation of a plane in $\R^3$

Parallel and Orthogonal Planes

Two distinct planes are parallel if their normal vectors are parallel (scalar multiples of each other)

• When two planes intersect, set $z=0$ to find a common point on both planes
  • Now we’ve simplified it to the intersection of two planes with the $xy$ plane
  • Direction vector of the line of intersection of the two planes can be found from the cross product of the two normal vectors

Two planes are orthogonal if their normal vectors are orthogonal (dot product of the two is zero)

13.6 Cylinders and Quadric Surfaces

Assigned questions: 1,13, 23, 29,31,33,35,37,49,60