Engineering Mechanics

Position Vector

Imagine you are standing at the main gate of your college and someone asks where the library is. You might say "40 metres straight ahead and 30 metres to the right." You just gave a position vector: a single instruction that points from a known starting place straight to the spot you care about.

In our last two lessons we learned to name points with Cartesian coordinates and to describe size-and-direction quantities with vectors. The position vector brings those two ideas together.

By the end of this lesson you will know what a position vector is, how to write it, how to find its length, and how to locate one point from another. This is the bridge between "where a point sits" and "the force acting there," which is why it matters so much in mechanics.

What Is a Position Vector?

A position vector is a vector that points from the origin to a specific point in space.

Remember the origin from our coordinate lesson? It is the fixed (0, 0, 0) starting point where the axes cross. A position vector is simply the arrow drawn from that origin straight to your point of interest.

So if a point sits at (4, 3), its position vector is an arrow starting at the origin and ending exactly at that point. The arrow answers one clear question: "starting from zero, how do I get to this point?"

This is the key feature that makes it special. An ordinary vector can float anywhere, but a position vector always begins at the origin. It is tied down, which is exactly what we need when we want to fix a location.

How We Write a Position Vector

A position vector is usually named r (for "radius" or "position"). Using the component form from our vector lesson, a point P at (x, y, z) has the position vector:

r = x i + y j + z k

  • r is the position vector of the point.

  • x, y, z are the coordinates of the point, measured in metres (m) in SI units.

  • i, j, k are the unit vectors pointing along the x, y, and z axes.

For a flat 2D case, we just drop the z term:

r = x i + y j

Here is the neat part. The numbers inside a position vector are simply the coordinates of the point. So the point (4, 3) has the position vector r = 4i + 3j. Coordinates and position vectors are two ways of saying the same thing, one as a location and one as an arrow.

How We Write a Position Vector

Magnitude of a Position Vector

The magnitude of a position vector is its length, which is the straight-line distance from the origin to the point.

We find it with the Pythagoras theorem, the same tool we used for the distance formula and for vector magnitude:

|r| = √(x² + y² + z²)

  • |r| is the length of the position vector, in metres (m).

  • x, y, z are the coordinates of the point.

In 2D, we leave out z:

|r| = √(x² + y²)

Solved Mini-Example

Find the magnitude of the position vector r = 4i + 3j.

  1. Square each component: (4)² = 16 and (3)² = 9

  2. Add them: 16 + 9 = 25

  3. Take the square root: √25 = 5

So |r| = 5 m. The point sits 5 metres away from the origin in a straight line. Notice this is exactly the distance from the origin to (4, 3), which confirms that a position vector's length is just its distance from zero.

The Position Vector Between Two Points

Points are rarely measured from the origin alone. Often you want the vector from one point to another. For example, from one bolt hole to the next, or from a joint to the point where a force is applied.

This is called the relative position vector, and it is the same as the displacement from the first point to the second.

For a vector pointing from point A(xA, yA, zA) to point B(xB, yB, zB):

r(AB) = (xB − xA) i + (yB − yA) j + (zB − zA) k

  • r(AB) is the position vector pointing from A to B.

  • The x, y, and z terms are each "the end coordinate minus the start coordinate."

The rule is simple: always subtract the starting point from the ending point. Head minus tail. If you reverse the order, the arrow points the opposite way.

Solved Mini-Example

Find the position vector from A(1, 2) to B(4, 6).

  1. Subtract the x values: 4 − 1 = 3

  2. Subtract the y values: 6 − 2 = 4

  3. Write it in component form: r(AB) = 3i + 4j

Its length is √(3² + 4²) = √25 = 5 m. So point B is 5 metres away from A, in the direction of that arrow. This "from one point to another" idea is used constantly when setting up forces along a member.

The Unit Vector Along a Position Vector

Sometimes you only care about the direction of a position vector, not its length. For that we use a unit vector, which has a magnitude of exactly 1 and points the same way.

We find it by dividing the position vector by its own magnitude:

r̂ = r / |r|

  • r̂ (r with a hat) is the unit vector along r.

  • r is the position vector, and |r| is its magnitude.

For r = 4i + 3j with |r| = 5:

r̂ = (4i + 3j) / 5 = 0.8i + 0.6j

This unit vector is pure direction. In mechanics we use it to take a known force size and point it along a rope, a link, or a member. You will see this trick often when a force acts along a slanted bar.

Position Vector vs Ordinary Vector

Both are vectors, so both have magnitude and direction. The difference is where they live:

  • A position vector always starts at the origin. Its whole job is to locate a point.

  • An ordinary (free) vector can start anywhere. It only describes a size and direction, like a force, and does not care where it sits.

Here is a simple way to picture it. A position vector is like the pin that drops on a map to show where you are. A free vector is like an arrow showing which way the wind is blowing, and that arrow means the same thing no matter where you draw it.

Where Engineers Use Position Vectors

Position vectors are not just neat maths. They do real work:

  • Locating forces on a body: To find the turning effect of a force (its moment, coming up soon), you need the position vector from the pivot to where the force acts.

  • CAD and 3D modelling: Every point, line, and feature in a model is stored as a position from the origin, which is the core idea behind tools like Fusion 360 and AutoCAD.

  • Robotics: A robot arm's gripper is tracked by its position vector, and reaching a target is a position-vector calculation, often done in Python.

  • Structural joints: Truss and frame problems place every joint using a position vector before any forces are solved.

Common Mistakes to Avoid

  • Forgetting the origin. A position vector must start at the origin. If it starts somewhere else, it is a relative position vector, not a plain one.

  • Subtracting in the wrong order. For the vector from A to B, always do B minus A. Reversing it flips the direction.

  • Mixing up the vector and its length. r is the full arrow; |r| is just its size.

  • Skipping units. Coordinates and magnitudes carry units. Keep them in metres and stay consistent.

Frequently Asked Questions

What is a position vector in simple words?

It is a vector drawn from the origin to a point. It tells you how to reach that point starting from zero.

What is the formula for a position vector?

For a point at (x, y, z), the position vector is r = xi + yj + zk. Its magnitude is √(x² + y² + z²).

What is the difference between a position vector and a displacement vector?

A position vector goes from the origin to a point. A displacement (relative position) vector goes from one point to another, found by subtracting the start coordinates from the end coordinates.

How do you find the magnitude of a position vector?

Use the Pythagoras theorem on its components: magnitude = √(x² + y² + z²). It equals the distance from the origin to the point.

Why are position vectors used in engineering mechanics?

They locate where forces act, which is essential for finding moments, solving joints, and modelling parts in CAD and robotics.

Key Takeaways

  • A position vector points from the origin to a specific point in space.

  • Its components are simply the coordinates of the point: r = xi + yj + zk.

  • Its magnitude, √(x² + y² + z²), equals the distance from the origin to the point.

  • The vector from A to B is found by subtracting A's coordinates from B's (head minus tail).

  • A unit vector along r gives pure direction and is found by r divided by |r|.

  • Position vectors locate where forces act, which is the base for finding moments.

Quick Revision Box

  • Position vector: r = x i + y j + z k

  • Magnitude: |r| = √(x² + y² + z²)

  • From A to B: r(AB) = (xB − xA) i + (yB − yA) j + (zB − zA) k

  • Unit vector: r̂ = r / |r|

Practice Corner

  1. Write the position vector of the point (5, 2, 3).

  2. Find the magnitude of the position vector r = 6i + 8j.

  3. Find the position vector from A(2, 1) to B(5, 5).

  4. What always marks the starting point of a position vector?

  5. What is the unit vector along r = 3i + 4j?

<details> <summary>Answers</summary>

  1. r = 5i + 2j + 3k.

  2. |r| = √(6² + 8²) = √(36 + 64) = √100 = 10 m.

  3. r(AB) = (5 − 2)i + (5 − 1)j = 3i + 4j.

  4. The origin, at (0, 0) or (0, 0, 0).

  5. |r| = √(3² + 4²) = 5, so r̂ = (3i + 4j) / 5 = 0.6i + 0.8j.

Conclusion

The position vector is how engineers say exactly where a point is, using an arrow from the origin. Once you can write it, measure its length, and find the vector from one point to another, you are ready to place forces precisely, which is the first step toward calculating their turning effect.

Want to practise until it clicks? Head to the Mechanical Engineer learning hub at GaugeHow and try the practice questions before the next lesson.

What's Next

Next up: the Unit Vector, where we focus purely on direction and learn how it lets us point any force exactly where we want it.

Internal links:

3D modelling in Fusion 360 → https://gaugehow.com/course/fusion-360

Cartesian coordinates in AutoCAD → https://gaugehow.com/course/autocad-2023

Python for Mechanical Engineers → https://gaugehow.com/course/python-for-mechanical-engineers-robotics

Mechanical Engineer Hub → https://gaugehow.com/mech

Practice tests → https://gaugehow.com/practice