Chapter 2 · Vectors CHAPTER 3 · PARTICLE EQUILIBRIUM →

Force Along a Line

In statics, forces often act along a cable, rod, link, or line connecting two points. A force along a line is written by combining a magnitude with a unit direction vector that points along the line of action.

Concept

A force vector can be expressed as force magnitude \(\times\) direction. The direction is provided by a units vector along the line of action.

Force Along a Line
Position Vector
\[ \mathbf{F} = F\,\hat{\mathbf{u}}_{AB} \]

The unit direction vector \(\hat{\mathbf{u}}_{AB}\) points from point \(A\) to point \(B\). It is formed from the displacement vector \(\mathbf{r}_{AB}\).

Procedure

Procedure — Force Along a Line

Procedure Vectors

Use two points on the line of action to build a displacement vector, normalize it, then multiply by the force magnitude.

  1. 1

    Select two points on the line

    Choose \(A(x_A,y_A,z_A)\) and \(B(x_B,y_B,z_B)\) on the force’s line of action.

  2. 2

    Form the displacement vector

    \[ \mathbf{r}_{AB}=\mathbf{r}_B-\mathbf{r}_A \]
  3. 3

    Compute its magnitude

    \[ \|\mathbf{r}_{AB}\|=\sqrt{(x_B-x_A)^2+(y_B-y_A)^2+(z_B-z_A)^2} \]
  4. 4

    Create a unit direction vector

    \[ \hat{\mathbf{u}}_{AB}=\frac{\mathbf{r}_{AB}}{\|\mathbf{r}_{AB}\|} \]
  5. 5

    Multiply by the force magnitude

    \[ \mathbf{F}=F\,\hat{\mathbf{u}}_{AB} =F\left(\frac{\mathbf{r}_{AB}}{\|\mathbf{r}_{AB}\|}\right) \]
Check: reversing the point order flips the direction: \(\hat{\mathbf{u}}_{BA}=-\hat{\mathbf{u}}_{AB}\).

Vector Form Using a Displacement Vector

Combining the previous steps gives a compact expression for a force along the line from \(A\) to \(B\):

Compact Form
\[ \mathbf{F} = F\,\hat{\mathbf{u}}_{AB} = F\left(\frac{\mathbf{r}_{AB}}{\|\mathbf{r}_{AB}\|}\right) \]

Here, \(\mathbf{r}_{AB}\) is the displacement vector pointing from \(A\) to \(B\), and \(\|\mathbf{r}_{AB}\|\) is its magnitude. The ratio \(\mathbf{r}_{AB}/\|\mathbf{r}_{AB}\|\) is a unit vector.

Common mistake. Do not divide by \(\mathbf{r}_{AB}\). You divide by the magnitude \(\|\mathbf{r}_{AB}\|\), which is a scalar.

Worked Example — Force Along a Line

Example Force Along a Line

Problem

A cable is pulled with 30 lb of force. Write the force that acts on point A as a Cartesian vector and determine the direction of the force. All lengths shown are in ft.

Cable from A to B with tension along the line
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