College Physics II – Mechanics, Sound, Oscillations, and Waves
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.
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The position vector originates from the origin (0,0,0) and terminates at the point (x,y,z).
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.
Position vectors are fundamental for describing motion in two and three dimensions.
The magnitude of a position vector is given by $|\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}$.
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.