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Position Vector

Position vector is used to help us find the location of one object relative to another object. Position vectors usually start at the origin and then terminate at any other arbitrary point. Thus, these vectors are used to determine the position of a particular point with reference to its origin.

In this article, let's learn about position vectors, their definition, formulas with solved examples.

1.	What Is a Position Vector?
2.	How To Find the Position Vector?
3.	Position Vector Formula
4.	FAQs on Position Vector

What Is a Position Vector?

The position vector is a straight line having one end fixed to a body and the other end attached to a moving point and is used to describe the position of the point relative to the body. As the point moves, the position vector will change in length or in direction or in both length and direction.

Position Vector Definition

A position vector is defined as a vector that indicates either the position or the location of any given point with respect to any arbitrary reference point like the origin. The direction of the position vector always points from the origin of that vector towards the given point.

In the cartesian coordinate system, if O is the origin and P(x₁, y₁) is another point, then the position vector that is being directed from the point O to the point P can be represented as OP.
In a three-dimensional space, if the origin O = (0,0,0) and P = (x₁, y₁, z₁), then the position vector v of point P can be represented as: v = x₁i + y₁j + z₁k

Let’s consider two vectors, P and Q, with position vectors p = (2,4) and q = (3, 5) respectively. The coordinates of the vectors P and Q can be written as: P = (2,4), Q = (3, 5). Let's consider an origin O as shown in the below image. We will consider a particle that moves from point P to point Q. The position vector of a particle can be defined as the vector that starts from the origin to the point where the particle is located.

Position Vector

In the above diagram, the position vector of the particle when it is at point P is the vector OP and when it is at point Q, it is OQ.

How To Find the Position Vector?

It's essential to first determine the coordinates of a point, before finding the position vector of that point. Consider two points, A and B, where A = (x₁, y₁) and B = (x₂, y₂).

Next, we will find the position vector from point A to point B, the vector AB.
To determine this position vector, we need to subtract the corresponding components of A from B: AB = (x₂ - x₁, y₂ - y₁) = (x₂ - x₁) i + (y₂ - y₁) j

Position Vector Formula

If we know the position of any point in the xy-plane, then we can use a formula to determine a position vector between those two points. For instance, consider a point A, which has the coordinates (x_k, y_k) in the xy-plane, and another point B, which has the coordinates (x_k+1, y_k+1).

The formula to determine the position vector from A to B is AB = (x_k+1 - x_k, y_k+1 - y_k).
The position vector AB refers to a vector that starts at point A and ends at point B.
Similarly, if we want to find the position vector from the point B to point A, then we can use: BA = (x_k - x_k+1, y_k - y_k+1)

Related Articles on Position Vector

Check out the following pages related to the position vector

Important Notes on Position Vector

Here is a list of a few points that should be remembered while studying position vector

A position vector is defined as a vector that indicates either the position or the location of any given point with respect to any arbitrary reference point like the origin.
The direction of the position vector always points from the origin of that vector towards the given point.

Examples on Position Vector

Example 1: Given two points P = (-4, 6) and Q = (5, 11), determine the position vector PQ.
Solution: If two points are given in the xy-coordinate system, then we can use the following formula to find the position vector PQ:

PQ = (x₂ - x₁, y₂ - y₁)

Where (x₁, y₁) represents the coordinates of point P and (x₂, y₂) represents the point Q coordinates. Thus, by simply putting the values of points P and Q in the above equation, we can find the position vector PQ:

PQ = (5-(-4), 11-6)

PQ = ((5+ 4), 11-6)

PQ = (9, 5) = 9 i + 5 j

Thus, the position vector PQ is equivalent to a vector that starts at the origin. This vector is directed to a point 9 units to the right along the x-axis, and 5 units upward along the y-axis.
Example 2: Given two points P = (-4, 6) and Q = (5, 11), determine the position vector QP.

Solution: If two points are given in the xy-coordinate system, then we can use the following formula to find the position vector QP:

QP = (x₁ - x₂, y₁ - y₂)

Where (x₁, y₁) represents the coordinates of point P and (x₂, y₂) represents the point Q coordinates. Note that the position vector QP represents a vector directed from point Q towards point P. It is different from the position vector PQ, which is directed from P to Q. Thus, by simply putting the values of points P and Q in the above equation, we can find the position vector QP:

QP = (-4-5, 6-11)

QP = (-9, -5) = -9 i - 5 j

Thus, the position vector QP is equivalent to a vector that starts at the origin. This vector is directed to a point of 9 units that is to the left along the x-axis, and 5 units downward along the y-axis.

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Practice Questions on Position Vector

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FAQs on Position Vector

What Is the Position Vector in Maths?

The position vector is a straight line having one end fixed to a body and the other end attached to a moving point and is used to describe the position of the point relative to the body. The direction of the position vector always points from the origin of that vector towards the given point.

How Do You Find the Position Vector?

It's essential to first determine the coordinates of a point, before finding the position vector of that point. Consider two points, A and B, where A = (x₁, y₁) and B = (x₂, y₂).

Next, we will find the position vector from point A to point B, the vector AB.
To determine this position vector, we need to subtract the corresponding components of A from B: AB = (x₂ - x₁, y₂ - y₁)

What Is the Difference Between a Position Vector and Displacement Vector?

A position vector is defined as a vector that indicates either the position or the location of any given point with respect to any arbitrary reference point like the origin. Whereas, the displacement vector helps us to find the change in the position vector of a given object.

Where Does a Position Vector Always Start?

The position vector starts at the origin and terminates at any other arbitrary point. These are used to determine the position of a point with reference to the origin.

What Is the Difference Between a Position Vector and a Unit Vector?

A vector is considered to be a unit vector when it is used to specify only the direction and has a magnitude equal to 1. There is no magnitude required for direction, hence the magnitude of the unit vector is always equal to 1. A position vector is defined as a vector that indicates either the position or the location of any given point with respect to any arbitrary reference point like the origin.

What Is the Formula for Position Vector?

Is Displacement a Position Vector?

No, a displacement is not a position vector. The position vector is used to help us find the location of one object relative to another object. It is essential to know the position of a body when we are describing its motion. But then, a displacement vector can be defined as the change or variation in the given position vector.

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