Vector Equations

Any linear system can be expressed as a single vector equation. For example this pair of equations

has the vector form

In this form we interpret the problem as follows. The unknowns x and y are coefficients applied to the constant vectors that appear on the left side of the equation. Clearly, as we vary the values of these coefficients, the resulting vector will also vary. The goal is to choose the x and y so that the resulting combination of vectors is just equal to the vector on the other side of the equation, namely, .

This sort of vector manipulation has a geometric interpretation. We visualize a vector such as as a directed line segment, drawn from the origin (0,0) to the point (4,-3). Multiplying a vector by a positive constant has the effect of expanding or contracting the length of the vector without changing its direction. So, for example, .8 is directed along the same line as , but is only .8 times as long. Similarly, multiplying a vector by a negative constant has two effects: it reverses the direction of the vector, and also expands or contracts the length.

When two vectors are added, the result can also be visualized geometrically. The original two vectors can be thought of as two sides of a parallelogram, with their common vertex at the origin. Then the sum of the two vectors appears as a line from the origin to the opposite vertex of the same parallelogram. This is the diagonal vector for the parallelogram.

With these geometric ideas, we now have a visual way to view the vector equation described above. As you try different values of x and y, the vectors and are stretched or compressed, (and possibly reversed). At the same time, the parallelogram defined by the two vectors will be modified as well. The point is to apply just the right combination of stretching/compressing/reversing to each vector, so that the resulting parallelogram has its diagonal vector exactly equal to .

This interaction is animated in the interaction window below. You define the vectors for the vector equation by typing numbers into the small colored boxes. Click a button to see the three vectors graphed. Now you can stretch/shrink/reverse two of the vectors (the red and blue ones from the left side of the equation), and try to make the diagonal vector for their parallelogram match the yellow vector on the other side of the equation. As you work with the buttons on the page, look for hints in the box below the numbered instructions.

Here are a couple variations on using the interaction above.

1. Parallel Vectors. Do you know how to identify parallel vectors? Two vectors are parallel when one is a multiple of the other. What happens to the vector equation if all the vectors are parallel? For example, consider an equation like this one:

2. Two Parallel Vectors. Now try a case where the two vectors on the left are parallel, but the one on the right is not. For example, try

3. Try another example where the vectors on the left side of the equation are not parallel, but the equation on the right side of the equation is parallel to one of the other two. Here is such an example: