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Position (geometry)

Position (geometry) is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Position (geometry) rather than just read about it. In short: In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point P in space. Its length represents the distance in relation to an arbitrary reference origin O, and its direction represents the angular orientation with respect to given reference axes.

Position (geometry) — main illustration
Position (geometry) — illustration

Key takeaways

  • Position (geometry) belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Position (geometry) to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Position (geometry) from memory before moving on to harder problems.

Reference excerpt

In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point P in space. Its length represents the distance in relation to an arbitrary reference origin O, and its direction represents the angular orientation with respect to given reference axes. Usually denoted x, r, or s, it corresponds to the straight line segment from O to P. In other words, it is the displacement or translation that maps the origin to P:

r = O P → . {\displaystyle \mathbf {r} ={\overrightarrow {OP}}.}

The term position vector is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus. Frequently this is used in two-dimensional or three-dimensional space, but can be easily generalized to Euclidean spaces and affine spaces of any dimension.

Relative position The relative position of a point Q with respect to point P is the Euclidean vector resulting from the subtraction of the two absolute position vectors (each with respect to the origin):

Δ r = s − r = P Q → , {\displaystyle \Delta \mathbf {r} =\mathbf {s} -\mathbf {r} ={\overrightarrow {PQ}},}

where s = O Q → {\displaystyle \mathbf {s} ={\overrightarrow {OQ}}} . The direction between two points is their relative position normalized as a unit vector.

Definition and representation

Three dimensions

In three dimensions, any set of three-dimensional coordinates and their corresponding basis vectors can be used to define the location of a point in space—whichever is the simplest for the task at hand may be used. Commonly, one uses the familiar Cartesian coordinate system, or sometimes spherical polar coordinates, or cylindrical coordinates:

… excerpt ends here. Continue reading the full article.

Illustrations

Position (geometry): Radius vector r → {\displaystyle {\vec {r}}} represents the position of a point P ( x , y , z ) {\displaystyle \mathrm {P} (x,y,z)} with respect to origin O. In Cartesian coordinate system r → = x e ^ x + y e ^ y + z e ^ z . {\displaystyle {\vec {r}}=x\,{\hat {e}}_{x}+y\,{\hat {e}}_{y}+z\,{\hat {e}}_{z}.}
Radius vector r → {\displaystyle {\vec {r}}} represents the position of a point P ( x , y , z ) {\displaystyle \mathrm {P} (x,y,z)} with respect to origin O. In Cartesian coordinate system r → = x e ^ x + y e ^ y + z e ^ z . {\displaystyle {\vec {r}}=x\,{\hat {e}}_{x}+y\,{\hat {e}}_{y}+z\,{\hat {e}}_{z}.}
Position (geometry): Space curve in 3D. The position vector r is parameterized by a scalar t. At r = a the red line is the tangent to the curve, and the blue plane is normal to the curve.
Space curve in 3D. The position vector r is parameterized by a scalar t. At r = a the red line is the tangent to the curve, and the blue plane is normal to the curve.
Position (geometry): Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a
Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a

Worked examples

Example 1 — a first encounter with Position (geometry)

Start with the simplest possible case. Write down what Position (geometry) claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Position (geometry) before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Position (geometry) ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Position (geometry)

In research
Position (geometry) appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Position (geometry) in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Position (geometry) is common in secondary-school and first-year university syllabi. It links to neighbouring topics Kinematic properties, Position, so understanding it makes those chapters shorter.
In everyday life
Look for Position (geometry) outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Position (geometry) in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Position (geometry) means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Position (geometry) out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Position (geometry) in simple terms?

In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point P in space. Its length represents the distance in relation to an arbitrary reference origin O, and its direction represents the angular orientation with respect…

Why does Position (geometry) matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Position (geometry)?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Position (geometry).

Tags

  • Kinematic properties
  • Position

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