Subsections

# Vector Computations and Vector Functions

## Purpose

The purpose of this lab is to use Maple to introduce you to a number of useful commands for working with vectors, including some applications. The commands come from the Maple linalg and CalcP7 packages which must be loaded before any of its commands can be used.

## Getting Started

To assist you, there is a worksheet associated with this lab that you can copy into your home directory by going to your computer's start menu and choose run. In the run field type:

\\filer\calclab


when you hit enter, you can then choose MA1023 and then choose the worksheet

Vector_start.mw


Remember to immediately save it in your own home directory. Once you've copied and saved the worksheet, read through the background on the internet and the background of the worksheet before starting the exercises.

## Background

The commands below are some of the most basic vector commands. Some examples using these commands can be found in the Getting Started Worksheet. More examples can be found in the Help screens for each command.

innerprod
Computes the dot product (also known as the inner product) of two vectors.
crossprod
Computes the cross product of two vectors.
evalm
Evaluates expressions involving vectors (and matrices).
norm
Computes the norm, or length, of a vector.

## Angle between two vectors

If is the angle between the vectors and , then .

## Vector Projection

The vector projection of onto or the component of in the direction of can be found using the following formula:

## Area of a Triangle

The triangle created by connecting the terminal ends of the vectors and in standard form has area = .

## Vector Functions

By parametric curve in the plane, we mean a pair of equations and for in some interval . A vector-valued function in the plane is a function that associates a vector in the plane with each value of in its domain. Such a vector valued function can always be written in component form as follows,

where and are functions defined on some interval . From our definition of a parametric curve, it should be clear that you can always associate a parametric curve with a vector-valued function by just considering the curve traced out by the head of the vector. The VPlot command is part of the plots package and will show the graph of the curve traced out by the terminal ends of the given vector function.

For this lab, we will assume that we have a vector-valued function that gives the position at time of a moving point in the plane. The velocity of this point is given by the derivative and the acceleration is given by the second derivative, . In many applications of curvilinear motion, we need to know the magnitude of the velocity, or the speed. This is easy to compute - just take the magnitude . If you think of the speed as the rate of change of distance along the curve, and recall that arc length is distance measured along the curve, then you have the following interpretation of the speed

where is arc length. If the speed is not zero for any value of in the interval , then it is possible to define a unit vector, that is tangent to the curve as follows.

Using this definition, you can write the velocity in the following form.

This is not the most useful form for calculating the velocity, but it does lead to a useful way of thinking about the acceleration experience by a particle moving in a curvilinear path. If the path is a straight line, acceleration depends only on whether the particle is speeding up or slowing down. In a curve, however, there is an additional acceleration, called the centripetal acceleration, that is needed to keep the particle moving on the curve. The magnitude of this acceleration depends on the speed of the car and how much the path is curving. It turns out that you can quantify this with an intrinsic property of the curve called the curvature, usually denoted , defined by the following equation.

That is, the curvature is the magnitude of the rate of change of the tangent vector with respect to arc length. For example, the curvature of a straight line is zero and it can be shown that the curvature of a circle of radius is the same for every point on the circle and is given by . The Maple Speed command computes the speed of a vector function and the Curvature command computes the curvature of a vector function.

## Exercises

1. Given the vectors and , and , use Maple to compute the numeric value of the following expressions, if possible. For those that cannot be computed because they make no sense, please explain what is wrong.

2. Find the angle between each of the vectors above.

3. Find a vector perpendicular to such that

where and . Then show that and are perpendicular.

4. Find the area of the triangle with vertices , and . (Hint: Shift the triangle to the origin by representing two of the sides of the triangles with vectors in standard form.)

5. The vector valued function represents uniform circular motion. Plot the vector function over the interval . Find the speed and curvature of . Why do you think it is called uniform circular motion?