Chapter 2 · Vectors CHAPTER 3 · PARTICLE EQUILIBRIUM →

Position Vectors

A position vector specifies the location of a point relative to an origin. In statics, position vectors are used to build displacement vectors between points and to form unit direction vectors that define lines of action in 2D and 3D.

Definition

A position vector is a vector that starts at the origin and ends at the point. It describes where the point is located by giving its signed distances along the coordinate axes.

Position Vector
\[ \mathbf{r}_P = \overrightarrow{OP} \]

Cartesian Form

In Cartesian coordinates, a position vector is written using unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\).

Position Vector (cartesian form)
Position Vector
\[ \mathbf{r} = r_x\,\mathbf{i} + r_y\,\mathbf{j} + r_z\,\mathbf{k} \]

Displacement Vectors Between Points

The displacement vector from point \(A\) to point \(B\) is obtained by subtracting the position vectors:

Distance Between Points
Position Vector
\[ \mathbf{r}_{AB} = \mathbf{r}_B - \mathbf{r}_A \]

In component form, if \(\mathbf{r}_A = x_A\mathbf{i}+y_A\mathbf{j}+z_A\mathbf{k}\) and \(\mathbf{r}_B = x_B\mathbf{i}+y_B\mathbf{j}+z_B\mathbf{k}\), then:

Component Form
\[ \mathbf{r}_{AB} = (x_B-x_A)\mathbf{i}+(y_B-y_A)\mathbf{j}+(z_B-z_A)\mathbf{k} \]

Unit Direction Vector

A unit direction vector tells you which way something points, but not how large it is. Its length is always 1. It is found by taking a displacement vector and dividing by its length so that only the direction remains. If you later multiply this unit vector by a magnitude, you get back a full vector with both direction and size.

Unit Direction Vector
\[ \hat{\mathbf{u}}_{AB}=\frac{\mathbf{r}_{AB}}{\|\mathbf{r}_{AB}\|} \]

The magnitude of the displacement vector is:

Distance Between Points
\[ \|\mathbf{r}_{AB}\|=\sqrt{(x_B-x_A)^2+(y_B-y_A)^2+(z_B-z_A)^2} \]
Common mistake. Don’t treat \(\mathbf{r}_{AB}\) as a unit vector. You must divide by \(\|\mathbf{r}_{AB}\|\) to obtain a direction-only vector.
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