Linear algebra part 2 - Plotting vectors

Plotting vectors on a graph is the best way to gain an intuative understanding of what they represent. Each axis of a graph allows us to show the magnitude (size) of one dimension. Stepping back from vectors for a minute, a simple number line allows you to show the size of a single number, one dimension. If we want to show the size of two numbers, a two-dimensional vector, we need two number lines, the x and y axis of a graph.

As an example let us define a two-dimensional vector, $\mathbf{v}$, and plot it on a graph. When we plot vectors the first component is plotted on the x-axis and the second component is plotted on the y-axis. The vector is shown as a straight line from $(0, 0)$ to this point.

Plotting vectors on a graph allows us to visualise adding and subtracting vectors. For example, we can define $\mathbf{w}$ and then calculate $\mathbf{v} + \mathbf{w}$.

The vector $\mathbf{v} + \mathbf{w}$ is the diagonal of the parallelogram formed by the vectors $\mathbf{v}$ and $\mathbf{w}$, that shares the same inital point as the two vectors (in this case $(0, 0)$). This is known as the parallelogram law of vector addition.

We see a similar picture when subtracting vectors. The vector $\mathbf{v} - \mathbf{w}$ is the diagonal of the parallelogram formed by the vectors $\mathbf{v}$ and $-\mathbf{w}$ that shares the same inital point as the two vectors (in this case $(0, 0)$).

Plotting vectors in three dimensions

Plotting a three dimensional vector requires three axes. The third component of the vector is plotted on the z-axis. We can define a three-dimensional vector, $\mathbf{v}$, and plot it and an x, y and z axis.

Adding and subtracting vectors in three dimensions works in the same way as two dimensions. For example, we can define a second vector, $\mathbf{w}$, and calculate $\mathbf{v} + \mathbf{w}$.

Problems

If you can solve these problems you have understood how to plot vectors. The following questions can all be answered on the same x and y axis.

1. Plot the vectors $\mathbf{v}$ and $\mathbf{w}$ on the graph, where:

2. Plot $\mathbf{v} + \mathbf{w}$.

3. Plot $\mathbf{v} - \mathbf{w}$.

Solutions