A vector can be expressed in unit vector notation by decomposing it into its component parts along the $x$, $y$, and $z$ axes.

To elaborate, a vector is a quantity that possesses both magnitude (size) and direction. In physics, we frequently encounter vectors in three-dimensional space, necessitating a standardized method of representation. This is where unit vector notation becomes essential.

Unit vectors are defined as vectors with a length of $1$ that point in the direction of the positive $x$, $y$, and $z$ axes. They are typically denoted by the letters $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$, respectively. For instance, $\mathbf{i}$ represents a unit vector in the direction of the positive $x$-axis, $\mathbf{j}$ in the direction of the positive $y$-axis, and $\mathbf{k}$ in the direction of the positive $z$-axis.

To express a vector in unit vector notation, one must decompose it into its components along the $x$, $y$, and $z$ axes. This decomposition is achieved by projecting the vector onto each axis. The magnitude of each projection represents the component of the vector along that particular axis.

For example, consider a vector $\mathbf{A}$ with components of $3$ along the $x$-axis, $-2$ along the $y$-axis, and $1$ along the $z$-axis. In unit vector notation, this vector can be expressed as:

Here, the coefficients of $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ (which are $3$, $-2$, and $1$, respectively) represent the components of the vector along the $x$, $y$, and $z$ axes.

It’s important to note that the sign of each component indicates the direction along its respective axis. A positive component implies that the vector points in the direction of the positive axis, while a negative component indicates that it points in the direction of the negative axis.

In conclusion, expressing a vector in unit vector notation involves breaking it down into its components along the $x$, $y$, and $z$ axes, and then representing it as the sum of these components multiplied by their corresponding unit vectors.