Introduction to the Algebra and Geometry of Euclidean Space

Sage Notebook: Vectors with Sage [.sws]
Dot products, cross products and magnitudes with Sage.
Sage Notebook: Lines and Planes with Sage [.pdf] [.sws]
Examples of finding Lines and Planes with Sage in 3D.
Vectors
Introduction to the concept of vector. Magnitud, direction, addition.
Vector Geometry
Cartesian and spherical coordinate systems. Describing, surfaces, lines, points with vectors.
Working with vectors in Maple
Using maple to compute addition of vectors, magnitudes, angles. The plane in the wind problem is here...
Simple Examples of Vectors with Maple
computing lenths, unit vectors, solving simple vector equations and the proof that the medians of a triangle intersect at a single point.
The Dot Product
Introducing the inner product. Scalar and vector projections.
The Cross Product
Definition. Cross products of the i,j,k basis vectors. Examples.
The Law of cosines
The famous law of cosines and the formula for the inner product in terms of the coordinates of the vectors.
Properties of Cross products
Maple proofs of the distributivity and anti-commutatitivity properties of cross products.
Cross products are NOT associative.
Maple proof that cross products are not associative.
Applications of the cross product: planes, volumes
Triple products. The volume generated by 3 vectors. Projected Area.
Lines with Maple
Position vector plus t times the velocity vector: Howto with maple.
The plane through 3 points
The equation of the plane containing 3 given points. The maple procedure P3points for computing it is here...
The plane containing two lines
The equation of the plane containing two given lines. The maple procedure interlines for finding the point of intersection of two lines in 3D is here...
The distance from a point to a line
How far away is this point from that line?
The maple proc d2line is here...
The distance from a point to a plane
How far is that point from this plane ?
The maple proc p2plane is here...
Plane containing two lines: Example1
Given two lines in symmetric form, maple is used to find the plane that contains them. A picture of the plane with the two lines is here...
Example: angle of diagonals
Simple Maple proof that when the diagonals of a rectangle intersect at right angles then the rectangle is a square.
Example: bisecting the angle between u and v
Length of u times v plus length of v times u does it! The proof with maple is here...
Two planes and one point
The equation of the plane that contains the line of intersection of two other planes and a given point.
Two planes, angle, line..
Finding the angle between two planes and the line of intersection in symmetric form.
A few review exercises
Seven problems on lines, planes, angles, innerprods etc...
More review exercises
Seven problems on planes, lines and vectors.

Carlos Rodriguez <carlos@math.albany.edu>