Difference Between Position Vector and Displacement Vector with Examples
Position vector and displacement vector are fundamental concepts in kinematics, which help in describing the motion and location of objects in a coordinate system. Understanding these vectors is essential for solving various problems in mechanics and other branches of physics.
Definition of Position Vector
A position vector is a vector that indicates the location of a point with respect to an origin in a coordinate system. It has both magnitude and direction, originating from the reference point (typically the origin) and ending at the given point in space.
In a two-dimensional plane, if a point A has coordinates $(x, y)$, its position vector is represented as $\vec{r} = x\hat{i} + y\hat{j}$. In three-dimensional space, for $A(x, y, z)$, $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$.
Position vectors are crucial for determining the relative location of bodies and for analysing motion related questions such as those in Kinematics Overview.
Formula for Position Vector Between Two Points
To find the position vector from point A $(x_1, y_1, z_1)$ to point B $(x_2, y_2, z_2)$, the formula is:
$\vec{AB} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}$
This vector represents the directed line segment from point A to point B in space. It is widely applied in topics such as Motion In 2D Dimensions.
Concept of Displacement Vector
The displacement vector is defined as the change in position vector of an object. It is a vector quantity that points from the initial position to the final position of the object, regardless of the path taken.
If an object moves from position $\vec{r}_i$ (initial) to position $\vec{r}_f$ (final), the displacement vector is given by:
$\vec{s} = \vec{r}_f - \vec{r}_i$
Displacement verifies both the magnitude and direction of the shortest route between two points, unlike distance which is a scalar.
Comparison: Position Vector and Displacement Vector
| Position Vector | Displacement Vector |
|---|---|
| Locates object from origin | Represents change from initial to final position |
| Examples: $\vec{r} = x\hat{i} + y\hat{j}$ | $\vec{s} = \vec{r}_f - \vec{r}_i$ |
| Origin-dependent | Path-independent |
| Magnitude: distance from origin | Magnitude: shortest distance between two points |
Example: Calculation of Displacement Vector
Consider a particle moves from point $A (1, 2, 3)$ to point $B (4, 6, 8)$. The position vectors of A and B are $\vec{r}_A = 1\hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{r}_B = 4\hat{i} + 6\hat{j} + 8\hat{k}$. The displacement vector is calculated as:
$\vec{s} = \vec{r}_B - \vec{r}_A = (4-1)\hat{i} + (6-2)\hat{j} + (8-3)\hat{k} = 3\hat{i} + 4\hat{j} + 5\hat{k}$
Properties of Position and Displacement Vectors
- Both are vector quantities (magnitude and direction)
- Position vector is measured from fixed reference
- Displacement vector is independent of path
- Displacement can be zero if initial and final positions coincide
- Magnitude of displacement ≤ actual distance travelled
Role in Kinematics
Position and displacement vectors are foundational for describing motion parameters such as velocity and acceleration. The change in position over time leads to velocity, while the change in velocity provides acceleration analysis, as addressed in Displacement Velocity And Acceleration.
These vectors also facilitate the study of relative motion, which is explained under Relative Velocity In Kinematics.
Frequent Applications and Related Areas
Applications of position and displacement vectors extend to projectile motion, analysis of movement in one and two dimensions, and graphical representation of motion. For specific applications, refer to Motion In One Dimension.
Mastery of these concepts is fundamental for solving kinematics problems and for advanced topics in physics and mathematics.
FAQs on Understanding Position Vector and Displacement Vector
1. What is a position vector?
Position vector is a vector that denotes the position of a point relative to a fixed origin.
- It is typically represented as r̅ or OP̅.
- In Cartesian coordinates, it is expressed as: r̅ = xi + yj + zk
- The position vector points from the origin to the specific point in space.
2. What is a displacement vector?
Displacement vector represents the change in position of an object from its initial to its final location.
- It is given by the difference: Δr̅ = r̅_{final} – r̅_{initial}
- Displacement is a vector quantity with both magnitude and direction.
- It is independent of the actual path taken.
3. How do you find the displacement vector between two points?
To find the displacement vector, subtract the initial position vector from the final position vector:
- Let initial position = r̅₁ = x₁i + y₁j + z₁k
- Let final position = r̅₂ = x₂i + y₂j + z₂k
- Displacement: Δr̅ = (x₂ - x₁)i + (y₂ - y₁)j + (z₂ - z₁)k
4. What is the difference between position vector and displacement vector?
Position vector specifies a location relative to the origin, while displacement vector measures the shortest straight-line change between two points.
- Position vector: From origin to a point
- Displacement vector: From initial to final position, regardless of the path
- Displacement can be zero even if a path is traveled (if starting and ending points coincide)
5. What are the properties of displacement vector?
Displacement vector has key properties that distinguish it as a vector quantity:
- Has both magnitude and direction
- Points from initial to final position
- Represents shortest distance (straight line) between two points
- Can be positive, negative, or zero depending on movement
6. Can the displacement vector be negative?
Yes, a displacement vector can be negative depending on the chosen direction.
- The sign indicates direction relative to the reference axis (positive or negative x, y, or z direction)
- Magnitude is always non-negative, but vector components may be negative
7. Is displacement a scalar or vector quantity?
Displacement is a vector quantity.
- It has both magnitude (distance) and direction
- It obeys vector addition rules
- Different from distance, which is a scalar
8. How is a position vector written in unit vector notation?
A position vector in unit vector notation is written as:
- r̅ = xi + yj + zk
- Where i, j, k are unit vectors along the x, y, and z axes respectively, and x, y, z are corresponding coordinates.
9. What is the formula for displacement in vector form?
The displacement vector is found using:
- Δr̅ = r̅_{final} – r̅_{initial}
- Expressed as: (x₂ - x₁)i + (y₂ - y₁)j + (z₂ - z₁)k
10. Does the displacement vector depend on the path taken?
No, the displacement vector only depends on the initial and final positions, not the actual path taken.
- It is the shortest straight-line distance between starting and ending points
- Total distance may vary, but displacement remains the same as long as endpoints are unchanged
