Different Representations of Vectors

Vectors are simple enough: they are objects that have a length and a direction in space. For example, my 3D position relative to the center of the earth is a vector because it has a length (my distance from the center), as well as a direction (my orientation with respect to the center). When I ride my bike, a vector can be used to describe the speed and direction in which I travel, it is called my velocity, \(\vec{v}\).

Temperature on the other hand is not a vector. Although temperature has a magnitude at every point, it has no direction, and the magnitude cannot serve as the length of anything. It is simply a number that is defined at every point in space (for the eager reader the technical term is a scalar field).

However, not every combination of length and direction is a vector. For example, we could say that \(\vec{b}_1\) represents the number of books I read while looking in the direction \(\theta = 0^\circ\), while \(\vec{b}_2\) represents the number of books I read while looking in the direction \(\theta=90^\circ\). However, if I read \(X\) books while facing \(\theta=0^\circ\) and then read \(Y\) books while facing \(\theta=90^\circ\), it is not equivalent to reading \(Z\) books while facing in some new direction. In fact, there is no way to combine these objects in a way that makes sense.

On the other hand, if I walk 1 m in the \(x\) direction, followed by 1 m in the \(y\) direction, it is clearly the same result as if I'd walked \(\sqrt{2}\) m in the direction \(\theta = 45^\circ\). There is a straightforward way to combine two displacements into a meaningful overall displacement. This property is essential for vectors in physics.

One of the common ways to represent a vector is to define it in terms of the Cartesian coordinate system. Let's agree to measure my East-West position by the \(x\) coordinate, and my North-South position by the \(y\) coordinate. For example, if I walk 1 m to the East of a sleeping elephant we can represent this motion by \(\vec{p}_1 =\langle 1, 0 \rangle\). If I then walk 1 m to his North, this motion can be represented by \(\vec{p}_2 = \langle 0, 1 \rangle\).

Vector addition

If this representation is consistent, we had better be able to represent my overall position relative to the sleeping elephant by the vector \(p_\textrm{total} = \langle 1, 1 \rangle\).

In fact, we see that to get the overall displacement, we may simply add the \(x\) and \(y\) entries of the two vectors, I.e.

\[\begin{align} \vec{p}_\textrm{total} &= \vec{p}_1 + \vec{p}_2 \\ &= \langle x_1, y_1 \rangle + \langle x_2, y_2 \rangle \\ &= \langle x_1 + x_2, y_1 + y_2 \rangle \\ &= \langle 1,1 \rangle, \end{align} \]

which is indeed equal to the overall position. This gives us a simple and consistent way to add vectors in the Cartesian representation. We simply add the vectors component-wise.

Another common representation for vectors is the polar coordinate system where we use \(\left(r,\theta\right)\) to specify our displacement relative to the elephant. For example, after \(\vec{p}_2\), when we're standing 1 m to the North of the sleeping elephant, we are at \(\vec{p}_2 = \left(1,90^\circ\right)\), which is to say that we stand a distance of 1 m from the origin at the angle \(\theta = 90^\circ\) to the origin. We see then that \(\langle 0,1\rangle\) and \(\left(1,90^\circ\right)\) are perfectly equivalent to one another.

Similarly we can represent \(\vec{p}_1 = \langle 1, 0\rangle\) by \(\left(1,0\right)\).

Calculating vector addition in polar coordinates

If we take both motions in order, we expect to arrive at \(\vec{p}_\textrm{total} = \big(\sqrt{2}, 45^\circ\big)\). Evidently, to compose vectors in the polar coordinate system, we can't simply add \(r_1+r_2\) to get \(r_\textrm{total}\) or \(\theta_1+\theta_2\) to obtain \(\theta_\textrm{total}\).

We recall the transformation between polar and Cartesian coordinates. If \(\langle x, y\rangle\) is the Cartesian representation of a vector, and \(\left(r,\theta\right)\) is the polar representation, it is the case that

\[\begin{align} x &= r\cos\theta \\ y &= r\sin\theta. \end{align} \]

Similarly, we can map back using the reverse transformation:

\[\begin{align} \theta &= \tan^{-1} \frac{y}{x} \\ r &= \sqrt{x^2 + y^2}. \end{align} \]

A convenient way to add the polar representations of vectors is to map into the Cartesian system, perform the addition, then map back into the polar system.

This holds for \(\vec{p}_\textrm{total}\):

\[\begin{align} \left(1,90^\circ\right) & \rightarrow \langle 0,1 \rangle \\ \left(1,0^\circ\right) &\rightarrow \langle 1,0 \rangle \\ \vec{p}_1 + \vec{p_2} &= \langle 1,1 \rangle. \end{align} \]

This is equivalent to

\[\begin{align} r &\rightarrow \sqrt{1^2+1^2} = \sqrt{2} \\ \theta &\rightarrow \tan^{-1} 1 = 45^\circ, \end{align} \]

yielding \(\big(\sqrt{2}, 45^\circ\big) \) as we expect.

One of the nice things about the Cartesian system is that vectors are added with ease, while it is a bit unnatural in the polar system. Each has its place however, and it is often much simpler to work in the polar system.

Despite the convenience of working in one representation or another, it is important to realize they are only tools for manipulation. The vectors exist independently and apart from the representations themselves. This means that the results of a vector calculation cannot depend on the coordinate system employed for the calculation, a fact that is extremely useful in circular motion, relativity, and other places later on in physics.