mathcomesalive

With ---VECTORS ALIVE--- you can explore the world of vectors - moving, resizing, linking, grouping, calculating and transforming then.

You are given 10 vectors to start with, a to j, and there are 16 more available.

Each vector has a start (or tail), a middle, and an end (or head):

                       ------------------o----------------->

                   start                    middle                  end

MOVING

Use the mouse pointer to grab (with the left button down) a vector by its start, and it can be moved by translation - same size, same direction, different position.


Grab it by its end (with the arrowhead) and you can move the end anywhere, but the start stays put.


Grab it by its middle and you can rotate the vector about its start, its length staying the same.  The vector may turn pink - ignore this for the moment.


LINKING

You can link vectors together, as a chain, linked head to tail, or as a bunch, all the tails or starts in the same place.

Move a vector into the linked position, or close by, and when you release the mouse button the vectors will become connected at that point.

A chain of vectors can be seen as a sequence of movements or displacements, or as a way of viewing the sum (see later) of a set of vectors.

A bunch of vectors can be seen as a set of forces acting at a point.


SELECTION

Most of the operations on vectors require you to specify which vectors are to be operated on.

You select a vector by clicking on its middle point without moving it. It will turn pink.

You can unselect it by clicking on the middle again.  You can clear all the selections by clicking on the background.


SUM

This is an operation on two or more vectors which produces a new vector, the sum, or what you get when you add them up. Select two vectors and click on "Sum".  A new vector is created, the sum.  With two separate vectors you can move them around and see that the triangle laww of addition works.

Try moving one of the blue vectors, and you will see the sum adjusted accordingly.


DIFFERENCE

This needs two vectors to be selected, and the second one is subtracted from the first one.

If the two vectors are linked at their start points, a mini bunch, then the difference vector will go from the end of the second vector to the end of the first.  this shows that if  d = b - a  then  b = a + d.


TRY THIS : Creates a chain of vectors, select them all and do SUM.  Moving any of the linking points of the chain has no effect on the sum.  Why not ?


ANGLE

A vector has size (length, magnitude) and direction, but each of these measurements needs a reference. However, we can find the angle between two vectors without reference values.

Select two vectors and click "Angle", and the angle will be displayed.  Counterclockwise is the positive direction for angles, and the angle value found is from the first selected to the second selected vector.

Move the vectors, the angle value is updated.


UNIT VECTORS

To make it possible to measure size and direction you can set up a unit vector.  Select any free vector and click on "Unit vector".  This vector is the first unit vector, and a second one is created at rightangles in the positive direction.  You can change the position, size and direction of the first unit vector; the second one follows.

Now when you select a vector its size and direction are shown, with some more details.


SCALAR MULTIPLES

A scalar multiple of a vector is a vector in the same direction whose size is equal to the size of the original vector multiplied by a scalar (number'.

First set up a value for scalar 'a' using the slider (you have already clicked on "Show scalars").

Then select a vector, and click on "Scalar mult".  The result appears slightly offset from the selected vector.


PROJECTIONS and COMPONENTS

A vector can be exressed as the sum of two parts, one in the direction of a given vector and one at rightangles to the given vector (draw a picture).  These can be found by a process of projection of the vector onto the given vector (for the first part) and a simple calculation (for the second part).

Select two vectors and click n "Projection".  You will obtain the projection of the second vector along the first vector.

When the given vector and the one at rightangles to it are a pair of unit vectors the amonts of each in the sum are called the components of the vector.  


DOT PRODUCTS (scalar products)

The dot product of two vectors is defined as the size of the first times the size of the second times the cosine of the angle between them,   a  dot  b  or  a.b   is size(a).size(b).cos(angle).

Select two vectors and click on "Dot product".  The dot product value is displayed.

The commonest simple use of the dot product is to determine whether two vectors are at rightangles or not.  How ???


POINTS and vectors

There are 20 labelled points A,...,T which can be moved around

and used to limit the freedom of 'free' vectors as follows:

1: Drag a point to a suitable position.

2: Select one or more free vectors.

3: Cloick on "Point at", "Ends at" or "Starts at".

4: Click on the point.

All the selected vectors will then do what it says.

Move the point and the vectors move as well, with the 'other' end fixed.

These vectors are no longer free, but for example a vector which starts at a point P can be made to end at a point Q.  One interesting situation where this is useful is a horizontal surface with three holes in it, and three strings joined at one end, with the other end passing through a hole, and equal weights on the other ends of the strings.

The following is the help file from VECTORS ALIVE, illustrated here with screen shots and video clips of the program in action.

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Selecting, moving and resizing vectors, then linking by their start points and finding the sum.

Linking vectors end to start, manipulating, moving the chain of vectors, and rotating the chain.

Selecting and resizing a vector, and then making it the unit vector. A unit vector at right angles is created, and the pair can be move resized and rotated.

Using the points.

Showing how vectors can be made to point at, start at or end at a point


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This video shows how the transformed vector l moves as the vector g is moved.

This video shows how the transformed vector l moves as the vector g is rotated around the origin.  The “Track” process shows the path followed.

LINEAR TRANSFORMATIONS (the hard stuff!)

Some background: A transformation (call it T) of vectors is a process which converts every vector into another one.  Where   a  is a vector we write  T(a)  for its transformation.

A linear transformation is a special type of process which converts any linear combination of vectors into the same linear combination of the converted vectors.  At which point we say thankyou to the mediaeval Arabs and do this in symbols:

 a  and  b  are vectors, T is the transformation, and are scalars (numbers). Then  

                           T( a  +  b ) = .T(a ) + .T( b )


Now, since in a two dimensional space any vector can be written as a linear combination of any two different vectors a linear transformation can be described exactly by its effect on a pair of different vectors.

Things become quite simple if the two different vectors are chosen to be the two unit vectors.


So, set up the linear transformation as follows:

1: Set up a unit vector (the other is created).

2: Take a free vector and place it with its start at the start of the unit vectors.

3: Move the end to a place of your choice.

4: Do steps 2 and 3 again with a second free vector.

These two vectors are to be the transformations of the unit vectors, and define the transformation.

5: Now take one more free vector, its start at the same place as the others, its end where you like.  This will be the vector which is transformed.

6: Select the three vectors, in the order thay were set up, and click on "Transform".

Now you can move your third vector around and see how its transform changes.

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Math Comes Alive

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