Position And Displacement Vectors

Position and displacement vectors play a role in mathematics, physics and engineering. They allow us to comprehend and describe the connections and dynamic movements of objects and points in three spaces. 

• These vectors are tools used across domains like computer graphics, animation, physics and engineering. 
• They help to determine positions, analyse shifts in location and distinguish between forces that cause motion, from those that maintain stability.

Key Terms: Vector, Displacement, Motion, Position Vector, Direction

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Position Vector 

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The concept of a position vector involves a line connecting an object to a moving point indicated by an arrowhead. 

• This vector is utilised to represent the points position in relation to the given object. 
• As the point moves, the length or direction (and sometimes both) of the position vector undergoes changes.

Characteristics of a Position Vector

The characteristics of Position Vector are mentioned below:

• A position vector defines the location of a point within coordinate space.
• It acts as a representation for pinpointing a point's location in three spaces.
• This vector finds application in computer graphics including areas such as animation, video games and visual effects.
• The position vector is essentially a three entity that specifies where an object resides within three dimensions.
• It has the capability to accurately describe any location.
• In addition to conveying location a position vector can also provide insights into an object's shape.

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Position Vector Formula

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The formulas of Position Vector are:

1. Position vector from point A to point B:

\(\vec{AB} \) = (x_k+1 – x_k, y_k+1 – y_k)

2. Position vector from point B to point A:

\(\vec{BA}\) = (x_k – x_k+1, y_k  – y_k+1)

Description of the formula

The formula is described as follows:

• The equations that describe position vectors in two dimensions exhibit a profound complexity. 
• The initial equation, wherein "AB" signifies the vector that denotes the position from point A to point B, encapsulates the very essence of this abstruse concept. 
• It is noteworthy that point A's x and y coordinates are symbolically represented by "x_k" and "y_k," respectively, while point B's coordinates take the form of "x_k+1" and "y_k+1." 
• The coordinates of A and B, the x and y components of the aforementioned vector "AB" can be effortlessly derived through a simple subtraction operation. 
• The second equation, commonly referred to as "BA," represents the position vector from B back to A and is acquired by deducting the x and y coordinates

Read More: Derivation of Equation of Motion in Physics

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Position Vector Example

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A position vector, through its mapping of a point on a plane to the corresponding location of the object it represents, assumes the role of an invaluable instrument capable of ascertaining the precise position of a particular point on a map. 

• Positioned primarily on the notion that the location rests at the centre of the map, these vectors possess the facility to align themselves harmoniously with a grid
• They expedite the determination of any object or point's placement on said map.

Position Vector

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What is a Displacement Vector?

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The Displacement vector shows how an item or person moves from one place to another in a dynamic way. It is found by taking the square root of the distance between two places in the opposite direction. It shows the path of the motion.

Characteristics of Displacement Vector

The characteristics are as follows:

• The displacement vector shows how an item or person moves from one point to another.
• The square root of the negative distance between these two points is used to figure it out.
• This trajectory mostly shows the direction of the movement, not how far it has gone.
• In physics, the displacement vector is one of the most important ideas used to show how a point changes over time.
• It helps tell the difference between the forces that move an item and those that keep it still.

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Displacement vector Formula

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The displacement vector represents how a point changes over time and is not affected by the laws of physics. It measures the distance between where the point is now and where it was when the last observation was made.

Equation:

To compute the displacement vector, we deploy the ensuing equation:

Displacement vector (s) = Final position (s_f) − Initial position (s_i)

Where:

• s signifies the transformative vector.
• s_i signifies the initial position of the point.
• s_f characterises the final position of the point.

Displacement Vector

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Things to Remember

• Position vectors are used in computer images, animation, and visual effects to show where a point is in three-dimensional space.
• Position vectors are found by taking the coordinates of one point and subtracting them from the coordinates of another point. 
• Position vectors give information about the shape and location of an item.
• Displacement vectors show dynamic movement between two places. 
• Displacement vectors are important in physics because they show the difference between forces that make things move and forces that keep things still.
• Displacement vectors are used to measure changes in position and have nothing to do with the physical rules that govern motion.
• They help people in different fields, like physics and engineering, understand the path and patterns of movement.

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Previous Year Questions

1. Extraction of metal from the ore cassiterite involves...[JEE Advanced 2011]
2. Commonly used vectors for human genome sequencing are...[NEET UG 2014]
3. Interfascicular cambium and cork cambium are formed due to​..
4. Pneumotaxic centre is present in​...[UP CPMT 2007]
5. Reaction of HBr with propene in the presence of peroxide gives….[NEET UG 2004]
6. Assuming the expression for the pressure exerted by the gas on the walls of the container, it can be shown that pressure is...[MHT CET 2016]
7. Which among the following is the strongest acid?...[TS EAMCET 2017]
8. Isopropyl alcohol on oxidation forms​..
9. A vector is not changed if​..
10. Which of the following arrangements does not represent the correct order of the property stated against it?...[JEE Main 2013]
11. The major product of the following reaction is​...[JEE Main 2019]
12. Major product of the following reaction is..[JEE Main 2023]
13. The percentage of nitrogen in urea is about..
14. The electric field at a point is​
15. Which of the following statements is true?​..[JKCET 2006]