Vector Components and Scalar Components

In math and physics we often have to deal with vectors in component form.

\(\vec{u} = 2\hat{i} - 3\hat{j}\)

This is a vector in component form. \(2\hat{i}\) and \(-3\hat{j}\) are its vector components, while the multiples of \(\hat{i}\) and \(\vec{j}\), i.e., 2 & -3, are its scalar components. 

Expressing a vector in component form basically means writing that vector in terms of some “base” vectors like \(\hat{i}\) and \(\hat{j}\). 

\(\hat{i}\) is a vector of length one unit in the positive x-direction; \(\hat{j}\) is a vector of length one unit in the positive y-direction.

Let’s say vector u’s tail is at the origin. Starting from the origin, we travel two times the length of \(\hat{i}\)\((2 \times 1 = 2)\) in the positive x-direction, and then travel 3 times the length of \(\hat{j}\)\((3 \times 1 = 3)\) in the negative y-direction(because the vector component is negative). We have reached the head of the vector u. 

So when a vector's tail is at the origin, the scalar components of the vector are the coordinates of the point where its head is located. This is a pretty common way to look at the components of a vector. 

Another way is to think of them as parts of the main vector acting along the two axes. They tell us “how much of that vector is acting along that axis”. Let’s look at another example of a vector defined from a point with coordinates (-3, 1) to the point (-2, 4). 

From the tip of its head and tail we drop perpendiculars onto the coordinate axes. The part of the vector acting along the x-axis(x-vector component) is of length (-2 - (-3) = 1), and in the positive x-direction. The y-vector component is of length (4 - 1 = 3) in the positive y-direction. So this vector can be written in component form as :

\(\vec{v} = 1\hat{i} + 3\hat{j}\)

Generally, if \((x_{h}, y_{h})\) and \((x_{t}, y_{t})\) are the coordinates of the head and the tail of a vector, the component form of the vector is then :

\((x_{h} - x_{t})\hat{i} + (y_{h} - y_{t})\hat{j}\)

We are subtracting x-coordinate of tail from x-coordinate of head for the component along x-axis, and subtracting y-coordinate of tail from y-coordinate of head for the component along y-axis.

The two base vectors \(\hat{i}\) and \(\hat{j}\) are also known as unit normal basis, since they are unit vectors, perpendicular(normal) and base vectors. Any vector in 2D can be written in terms of these two vectors. 

However, these two aren’t the only pair of base vectors. Infact, any two non-parallel vectors in 2D can be chosen as base vectors! If \(\vec{a}\) and \(\vec{b}\) are non-parallel vectors then 

\(\vec{c} = \alpha\vec{a} + \beta\vec{b}\)

spans the entire 2-dimensional space! It means, for any \(\vec{d}\) in 2D there will always exist unique \(\alpha\) and \(\beta\) such that \(\vec{c}\) and \(\vec{d}\) are equal. 

But in most earlier introductory courses on vectors we only ever see \(\hat{i}\) and \(\hat{j}\) as base vectors. There is however one application where any two non-parallel vectors can be used as base vectors, and is that for finding the parametric equation of a plane in 3D.

Know more - Parametric equation of a plane in 3D