Given two points in two- or three-dimensional space:
Calculate the distance between them and their midpoint.
Determine a parameterization of the line that contains those points.
Given two vectors in two- or three-dimensional space:
Calculate and sketch any linear combination of them.
Calculate the measure of the angle formed between them.
Calculate the vector that results from projecting one onto the other.
Calculate a vector that is orthogonal to both of them.
Given three non-colinear points in space,
determine an equation for the plane that contains those points.
Problems
What is the shortest distance
from the point \((-4,7)\) to the line \(3x-4y=10?\)
In general, determine a formula for the shortest distance
between a point \((p,q)\) and a line \(Ax+By=C\)
in terms of \(p\) and \(q\) and \(A\) and \(B\) and \(C.\)
[now in 3d]
What is the shortest distance
from the point \((TK)\) to the plane \(TK?\)
In general, determine a formula for the shortest distance
between a point \((p,q,r)\) and a plane \(Ax+By+Cz=D\)
in terms of \(p\) and \(q\) and \(r\)
and \(A\) and \(B\) and \(C\) and \(D.\)
Given a line in three dimensional space
and a point not on that line,
compute the distance from the point to that line.
Given two non-intersecting lines in space
compute the closest distance between them.
Given a plane in space and a line not parallel to that plane,
determine the point at which the line intersects that plane.
Given two non-parallel planes in space,
determine a parameterization of the line along which they intersect.
Suppose you’re swimming in a river with a current.
What direction do you swim in to get across?
[same but with a plane working against wind]
Mr goat, but he can fly.
3d analog of the pipe problem from /synthetic
Any two diagonals of a cube intersect in the center of the cube.
What is the acute angle at which two diagonals of a cube intersect?
TK Scalar triple product is volume of parallelepiped.