8.4 Visualization of the Basic Operations for Vectors

The basic operations for vectors, i.e. summation, subtraction, inner and outer product, that have been sketched above, are visualized in the following interactive three dimensional simulation Fig.8.2. This simulation starts by creating two randomly orientated position vectors (zero-point vectors) $\mathbf{a}$ and $\mathbf{b}$ of length 1, that are embedded in a transparent sphere of radius 1. In the figure the summation of these two vectors is shown.

The Orientation of the axes in space can be adjusted with the mouse and will be perceived as rotation of the sphere. Every activation of the button new vectors creates a new pair of vectors. The coordinates of these vectors are shown on the left.

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Figure 8.2: 3D-presentation of vectors, their sum, difference, outer product and multiple sum. The vector coordinates are shown on the right, the angle between the vectors, their inner product and the absolute value of the outer product on the bottom. On the left different projections can be chosen. The Figure shows the perspective representation of the vector sum $\mathbf{a}+\mathbf{b}$. Pushing the button new vectors creates randomly orientated vectors.

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Using the option switches on the left different well-defined viewing projections can be selected.

Next to the vector switch the angle between the vectors, the product of their absolute values (here always 1 because of normalization), the scalar product and the absolute value of the vector product are displayed.

With the option switches the different vector operations are visualized and superpositions are possible.

For addition and subtraction the input vectors are complemented by lines related through parallel translation. This visualizes the construction of the red result vector from the parallelograms.

$\mathbf{a}\times \mathbf{b}$ creates the vector product $\mathbf{a}$ cross $\mathbf{b}$ and displays it as black arrow. If the sphere is rotated in such a way , that the plane defined by the two input vectors lies in the figure plane, then these vectors just touch the equator of the sphere and one looks along the direction of the resulting vector. This demonstrates the orthogonal direction. If one moves with the right hand from $\mathbf{a}$ via $\mathbf{a}$ to the vector product one completes a right handed screw.

Performing the same experiment with $\mathbf{b}\times \mathbf{a}$ one completes a left handed screw. This is the meaning of the non-non-commutativity of the vector product: the direction of the vector product $\mathbf{b}\times \mathbf{a}$ is opposite to that of $\mathbf{a}\times \mathbf{b}$, thus we have $\mathbf{b}\times \mathbf{a}=-\mathbf{a}\times \mathbf{b}$. If one displays $\mathbf{a}\times \mathbf{b}$ next to $\mathbf{b}\times \mathbf{a}$, one sees, that both vectors have the same length, but point in opposite directions.

Finally $\mathbf{a}+\mathbf{b}+\mathbf{c}$ creates three random vectors and their red sum vector. If one activates $\mathbf{a}+\mathbf{b}$ in addition, one recognizes the partial construction of the sum of the first two vectors and one can implement the completion to the total sum vector in ones imagination.

In the description of the simulation you find further details and suggestions for experiments.