Standard Unit Vectors: $\underset{\sim}{i}$ and $\underset{\sim}{j}$

When giving directions to someone, we often use the compass points 'north', 'east', 'south' and 'west' rather than an exact direction like '34 degrees north of east'. This system works because we are able to create an L-shaped path which ends up in the same location. In fact, we are able to do this for any point on the map using just the directions north (or south) and east (or west).

Imagine we are working with some vectors on a map. If we have a vector pointing north and a vector pointing east then we can recreate any north-east vector using scalar multiplication of these two vectors and then adding them. Further, because we can use negative scalar multiplication to produce vectors pointing south or west, we can recreate any vector in any direction using just these two vectors.

Since the original length of these vectors doesn't really matter (we can just scale it to whatever we would like), we could just work with the vectors point north and east that have a convenient length (like 1km, or 1mm, but this depends on the context of our work). We recognise the special status these two vectors as being the 'standard units' which can construct any vector we are working with, and so we call these the standard unit vectors of our map.

In two-dimensional problems, we often call these $\underset{\sim}{i}$ and $\underset{\sim}{j}$ (for the unit vectors along the $x$-axis and $y$-axis respectively), and in three-dimensional problems we also include $\underset{\sim}{k}$ which is a unit vector pointing along the $z$-axis. When we break up vectors into components based on these standardised unit vectors, we make it easier to communicate and construct vectors in a diagram.

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