Position vector

Definition

A position vector is a vector that specifies the position of a point in space relative to an origin. It is represented as $\mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}$ in three dimensions.

5 Must Know Facts For Your Next Test

1. The position vector originates from the origin (0,0,0) and terminates at the point (x,y,z).
2. It can be expressed in Cartesian coordinates as $\mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}$ or in polar coordinates depending on context.
3. Position vectors are fundamental for describing motion in two and three dimensions.
4. The magnitude of a position vector is given by $|\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}$.
5. Position vectors are used to calculate displacement and velocity vectors by differentiating with respect to time.

Review Questions

• How do you express a position vector in Cartesian coordinates?
• What is the magnitude of the position vector $\mathbf{r} = 3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}$?
• Explain how a position vector differs from a displacement vector.

Related terms

Displacement Vector:

A vector that represents the change in position from one point to another. It is given by $\Delta \mathbf{r} = \mathbf{r}_f - \mathbf{r}_i$ where $\mathbf{r}_f$ and $\mathbf{r}_i$ are the final and initial position vectors.

Velocity Vector:

A vector that represents the rate of change of displacement with respect to time. It is defined as $\mathbf{v} = \frac{d\mathbf{r}}{dt}$.

Cartesian Coordinates:

A coordinate system that specifies each point uniquely by a set of numerical coordinates, which are the signed distances from the point to two or three fixed perpendicular directed lines, measured in the same unit of length.