Chapter 1: Vectors, Lines and Planes
Section 1.1 - Cartesian Coordinates and Vectors
Section 1.2 - Vector Arithmetic
Section 1.3 - Dot Product
Section 1.4 - Cross Product
Section 1.5 - Applications of Vector Products
Section 1.6 - Lines
Section 1.7 - Planes
The typical multivariate-calculus course begins with activities designed to help the student begin to think in three dimensions. The Cartesian coordinate system for both two and three dimensions is articulated, and the notion of the "position vector," an arrow from the origin to a Cartesian point is defined in terms of the unit basis vectors i,j,k.
Various notations are in use for vectors. This ebook uses a bold Roman font, and eschews the use of surmounted arrows, as in V→, because assigning to such a symbol in Maple would require that it be made into an Atomic Variable, an added (and tedious) step each time the symbol is invoked.
In Cartesian coordinates, the position vector and the point at its head can be (and are) identified. Some texts introduce the term bound vectors to denote vectors whose "tail" is attached to a specified point. Thus, in Cartesian space, the notion of parallel transport of a vector is implicit.
Once vectors have been introduced, their manipulation and use is explored. In this regard, the dot, cross, and box products appear, and are used in the calculation of area, volume, and projections.
A study of lines and planes in three-dimensional Cartesian space completes the introduction to visualization in 3D, and the interaction of line, planes, and vectors provides a wide field of exercises that test the student's ability to think about these linear objects.
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