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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.
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Working with vectors in Maple
- Using maple to compute addition of vectors, magnitudes, angles.
The plane in the wind problem is here...
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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.
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The Cross Product
- Definition. Cross products of the i,j,k basis vectors.
Examples.
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The Law of cosines
- The famous law of cosines and the formula for the inner product
in terms of the coordinates of the vectors.
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Properties of Cross products
- Maple proofs of the distributivity and anti-commutatitivity
properties of cross products.
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Cross products are NOT associative.
- Maple proof that cross products are not associative.
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Applications of the cross product: planes, volumes
- Triple products. The volume generated by 3 vectors. Projected Area.
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Lines with Maple
- Position vector plus t times the velocity vector: Howto with maple.
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The plane through 3 points
- The equation of the plane containing 3 given points. The maple
procedure P3points for computing it is here...
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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...
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The distance from a point to a line
- How far away is this point from that line?
The maple proc d2line is here...
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The distance from a point to a plane
- How far is that point from this plane ?
The maple proc p2plane is here...
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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...
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Example: angle of diagonals
- Simple Maple proof that when the diagonals of a rectangle
intersect at right angles then the rectangle is a square.
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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...
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Two planes and one point
- The equation of the plane that contains the line of intersection
of two other planes and a given point.
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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...
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More review exercises
- Seven problems on planes, lines and vectors.
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