Describing direction of a Vector in 2D

Describing Direction of a vector in 2D

(If you want to read a post on how to describe the direction of a vector in 3D which also includes Direction angles and direction cosines, click here) 

There are many ways to specify the direction of a vector in 2D, depending on what information we are given about it. Let \(\vec{v}\) be a vector given in component form, 

\(\vec{v} = 2\hat{i} + 2\hat{j}\)

We could simply say that \(\vec{v}\) is pointing somewhere in north-east direction to kinda roughly describe the direction. But why give a sketchy account of direction when there are simple and better ways which allow us to precisely describe the direction for a vector given in component form. 

One way to perfectly describe the direction of a vector, is from its two scalar components, 2 and 2. If we choose origin as the tail of the vector arrow representing \(\vec{v}\), then the coordinates of the head will be the X and Y scalar components. 

We can specify the direction of \(\vec{v}\) as : 2 units in positive X-direction & 2 units in positive Y-direction. If you verbally share this description to your friend, she would be able to easily locate the direction of the vector on 2D plane. She would also be able to determine the magnitude of the vector(length of the arrow) from its given scalar components. 

We could also use the terms ‘east’ and ‘north’ for ‘right’ and ‘up’; 2 units east & 3 units north. 

Similarly, if \(\vec{p} = -3\hat{i} - 5\hat{j}\) 

is another vector in this 2D space, its direction can be expressed as : 3 units in negative X-direction & 5 units in negative Y-direction. 

If we don’t know the scalar components of a vector, there is another way to specify the direction of a vector by stating the angle the vector makes with either the horizontal or the vertical axis. For instance, in the example of \(\vec{v}\), we could say that it forms an angle 45° with the X-axis.  

As you can tell from the figure on the left, merely stating the angle with an axis is not going to be sufficient information. There are four directions in 2D that make 45° angle with X-axis and \(\vec{v}\) could be in either of those four directions. 

So what should we do? 

We can bring down the four possible cases to two by stating the side of the X-axis with which the 45° angle is formed. For example we could say, 45° with the positive arm of X-axis. 

So \(\vec{v}\) could be in either of the two directions that form 45° angle with positive X-axis. We still need to provide more information to further reduce it to the only correct possible result. We see in the above figure that one of the two angles is in counterclockwise direction starting from the positive arm of the X-axis, and the other angle starting from the positive arm of X-axis is in clockwise direction. 

So to further reduce it to the only correct case, we can specify the direction of the angle. We could say, the direction of \(\vec{v}\) is 45° counterclockwise from the positive arm of the X-axis. This is a complete description for the direction of \(\vec{v}\) as it accurately expresses its direction. If you’d share this description with your friend, she would be able to easily locate the direction of the vector in 2D space, but this time she will not know what length the arrow will possess, because this description of direction doesn’t really tell her anything about the magnitude of \(\vec{v}\). 

To completely describe the vector, both in magnitude and direction, you could say, 4 units 45° counterclockwise from the positive arm of the X-axis, where 4 units is magnitude of \(\vec{v}\).  

In geographical terms, we could also describe this direction as 45° north of east. 

All good so far, but we should be very careful about one thing regarding the above description of direction. The 45° angle is counterclockwise from positive X-axis, and not 45° counterclockwise from the vector. The latter will lead us to the wrong case. In other words, under the basic convention, the arc of an angle should start from the positive arm of an X-axis in counterclockwise direction, and end at the vector. 

Similarly, the direction of (\vec{u}\) can be described as, 239.036243° counterclockwise from the positive arm of the X-axis, which practically can be rounded about to 239°. 

Even though the above angle description for direction of a vector is proper and complete, it is not unique. We could also have specified the direction of \(\vec{v}\) as 315° clockwise from the positive arm of X-axis, or 225° clockwise from negative arm of Y-axis. Each of the two can also properly and completely describe the direction of \(\vec{v}\). 

It is our choice to choose any fixed direction in 2D and describe any angle clockwise or counterclockwise from that fixed direction. But it is better to stick with just one set of rules especially if there are multiple vectors involved. In many texts written on the topic, you will often find the direction to be measured from the positive arm of the X-axis in counterclockwise direction.