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 dT/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 

a =   dv dt  T + κ v2 N

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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