Position Vectors and Displacement Vectors: We came across upward or downward motion, as well as a motion to the left or right of the previously chosen origin, while studying one-dimensional motion. We also gave a positive sign to the right (or upward) and a negative sign to the left (or downward motion). We discussed positive and negative displacements, positive and negative velocities, and positive and negative accelerations in this context.
As a result, in one-dimensional motion, the concept of direction was constrained, and we could only discuss it using positive and negative numbers for displacement, velocity, and acceleration. When describing two- or three-dimensional motions, however, the concept of direction takes on a new meaning.
This is owing to the fact that a particle can have displacement, velocity, or acceleration in multiple directions in two and three dimensions. The concept of vectors must be introduced to deal with such a circumstance. With that said let’s begin with today’s topic which is
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In this article we learned, the position vector of an object at time t is the object’s position in relation to the origin. A straight line connecting the origin and the position at time t is used to depict it. The displacement vector of an item between two positions is the straight line that connects the two points, regardless of the path taken.
The displacement is always equal to or larger than the path length. Stay tuned for more such content ahead! We came across upward or downward motion, as well as motion to the left or right of the previously chosen origin, while studying one-dimensional motion. We also gave a positive sign to the right (or upward) and a negative sign to the left (or downward motion). We discussed positive and negative displacements, positive and negative velocities, and positive and negative accelerations in this context.
Position vectors represent the location of a moving point relative to a body by using a straight line with one end attached to it and the other to the body. The position vector will change length, direction, or both length and direction as the point moves.
When a position vector is drawn to a scale, a change in length indicates a change in magnitude, while a change in direction indicates a rotation of the vector. A position vector can only experience changes in magnitude and direction, and the velocity of the point is defined as the time rate of change of the position vector.
In the picture above, the particle’s position vector when it is at point P is OP, and when it is at point Q is OQ.
Displacement Vector
The displacement vector is defined as the change in an object’s position vector.
Only the starting and ending position vectors affect the displacement vector. When an object follows a path and returns to its original position, the displacement is regarded to be zero.
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We must first identify the coordinates of a point before we can determine its position vector. Let’s say we have two points, M and N
And let’s say M and N have (x1, y1) (x2, y2) as their coordinates respectively
And let’s say we are required to find position vector from point M to the point N
We subtract the corresponding components of M from N to get this position vector:
Therefore MN=(x2–x1), (y2–y1)
Let us have a more clear understanding of position and displacement vector by looking into some examples
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Position and Displacement of vectors examples (Solved)
Question 1: Determine the position vector AB given two points A = (-3, 4) and B = (6, 12). Determine the magnitude of vector AB.
Solution: We need to find position vector AB
We know the formula, MN=(x2–x1), (y2–y1)
Therefore, AB=[6-(-3)], (12-4)
MN = 9, 8
Now Magnitude of MN= 92+82
81+64=145
Question 2: Determine the position vector PQ given two points P = (-2, -4) and Q = (6, -2). Find the magnitude of the vector AB.
Solution: We need to find position vector AB
We know the formula, MN=(x2–x1), (y2–y1)
Therefore, PQ=[6-(-2)], [(-2-(-4)]
PQ = 8, 2
Now Magnitude of PQ= 82+22
64+4=68
Question 3: The position vector of a particle traveling in a plane is given as r = x2i+y3j
Calculate the displacement between x = 1 and x = 4 seconds.
Solution: As mentioned above, Displacement only depends upon initial and final position
Therefore r = x2i+y3j at x = 1
= i+j
And similarly r = x2i+y3j at x = 4
=16i+64j
The difference in their position vectors will determine the displacement between them.
Displacement = rfinal–rinitial
(16i+64j)-(i+j)
15i-63j
Question 4: The position vector of a particle traveling in a plane is given as r = x3i+y4j
Calculate the displacement between x = 2 and x = 6 seconds.
Solution: As mentioned above, Displacement only depends upon initial and final position
Therefore r = x3i+y4j at x = 2
= 8i+16j
And similarly r = x3i+y4j at x = 6
=216i+1296j
The difference in their position vectors will determine the displacement between them.
Displacement = rfinal-rinitial
(216i+1296j)-(8i+16j)
208i-1280j
Position Vector: Describes the position of a point in space relative to an origin. Displacement Vector: Describes the change in position of an object, indicating how far and in what direction the object has moved from its initial position to its final position. In essence, the position vector describes a location, while the displacement vector describes a movement or change in position.
In physics, position vectors and displacement vectors are used to describe the motion of objects: The position vector describes the location of an object at a given point in time. The displacement vector describes how far an object has moved and in which direction, from its initial position to its final position. This is crucial for understanding the path and distance traveled by an object during motion.
In navigation, position vectors are used to represent the location of an object or vehicle relative to a reference point, such as a starting point, origin, or coordinate system. The position vector is used to track the movement and direction of the object. Displacement vectors help determine the distance and direction the object has traveled during its journey.