# Vectors

### Summary and Review

We began this chapter exploring the concept of a *vector*,
along with the vector operations of the *dot product* and the
**cross product. **The dot product is used to define the *length*
of a vector, along with the *angle* between two vectors.
Indeed, one of the most important ideas introduced in this chapter is
that two vectors are at right angles to one another only if their dot
product is zero. The concept of a *projection* of a
vector onto a non-zero vector is based on this important concept, as is
the *point-normal* equation of a plane. The cross product of
two vectors **u** and **v**, on the other hand, produces a new
vector that is orthogonal to both **u** and **v**, and it is this
concept which allows us to produce normal vectors to planes. The **magnitude
of the cross product **of two vectors is equal to the area of the
parallelogram they determine.

The operations and arithmetic of
vectors allow us to study curves, in that if **r**(*t*) is a *position
vector* for each time *t, *which is to say that it begins at
the origin, then the endpoint of **r**(*t*) over [*a,b*]
traces out an *oriented curve. * The *vector-valued
function *r(*t*) is subsequently called a *parameterization
*of the curve. The derivative **v**(*t*) is called the *velocity*
of the curve parameterized by **r**(*t*) and is tangent to the
curve at the point corresponding to the end of **r**(*t*).
The length of **v**(*t*) is the *speed* of the curve
and is denoted *v*(*t*) or *ds/dt. *The integral of
the speed *v*(*t*) over [*a,b*] is the distance traveled
along the curve from "time" *t=a* to time *t=b, *which
is known as the *arclength * of the curve. The **unit
tangent **vector **T**(*t*) is the unit vector in the
direction of the velocity **v**(*t*) and the **unit normal **vector
**N**(*t*) is the unit vector orthogonal to **T**(*t*)
and in the direction of the derivative *d***T***/**dt.
*The *unit binormal *vector **B**(*t*) is the
cross product of **T**(*t*) and **N**(*t*).
Finally, the *curvature* κ(*t*) = 1/*R*(*t*)*,
*where *R*(*t*) is the radius of the "best
approximating circle" or *osculating circle *, to the
curve with center on a line parallel to **N**(*t*). The
curvature is important because it allows us to reduce the derivative of
the velocity, which is known as the *acceleration *of the
curve and is denoted **a**(*t*), to the physically meaningful
representation

There are other ideas and concepts introduced in this chapter, and you should
re-read the individual sections in addition to this summary. The
review materials are based on both the ideas above and some of those in
the chapter not mentioned here. Review questions and solutions are
in web page form on the left and in pdf form on the right. For
maximum benefit, you should attempt to answer the questions *before*
you look at the solutions.

You will need Acrobat reader in order to open and print the pdf files.

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