How do you represent vector f using unit vectors?

You represent vector **f** using unit vectors by expressing it as a combination of its components along the x, y, and z axes.

In more detail, a vector can be broken down into its components along the coordinate axes. For a three-dimensional vector **f**, these components are along the x, y, and z axes. Each of these axes has a corresponding unit vector: **i** for the x-axis, **j** for the y-axis, and **k** for the z-axis. A unit vector has a magnitude of 1 and points in the direction of the axis it represents.

To represent vector **f** using unit vectors, you write it as a sum of its components multiplied by the corresponding unit vectors. For example, if vector **f** has components \(a\), \(b\), and \(c\) along the x, y, and z axes respectively, you can write:

\[ \mathbf{f} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \]

Here, \(a\), \(b\), and \(c\) are the magnitudes of the vector's components along the x, y, and z axes. The unit vectors **i**, **j**, and **k** ensure that each component is correctly oriented along its respective axis.

For a two-dimensional vector, you only need the x and y components, so you would write:

\[ \mathbf{f} = a\mathbf{i} + b\mathbf{j} \]

This method of representing vectors is very useful in physics and engineering because it clearly shows how much of the vector points in each direction. It also makes it easier to perform vector operations like addition, subtraction, and scalar multiplication.