A position vector, written r⃗, is the vector from the origin of your coordinate system to a particle's location. In AP Physics C: Mechanics, r⃗(t) is the starting point for kinematics (differentiate it to get velocity and acceleration) and for torque (τ⃗ = r⃗ × F⃗).
A position vector r⃗ answers one question. Where is the object right now, measured from the origin? In components, r⃗ = x î + y ĵ + z k̂, and its magnitude |r⃗| = √(x² + y² + z²) is the straight-line distance from the origin to the object. The vector itself has both magnitude and direction, which is what makes it a vector and not a scalar (Topic 1.1).
In Physics C, the position vector almost always shows up as a function of time, r⃗(t). That one function contains the particle's entire motion. Differentiate it once and you get the velocity vector v⃗ = dr⃗/dt. Differentiate again and you get acceleration a⃗ = dv⃗/dt. A vector like r⃗(t) = (3cos ωt) î + (3sin ωt) ĵ describes a circle of radius 3; add a vt k̂ term and you get a helix. Reading the geometry of a motion straight out of r⃗(t) is a core Physics C skill.
The position vector lives in Topic 1.1 (Scalars and Vectors), where it's your first real example of a vector quantity, and it comes roaring back in Topic 5.3 (Torque), where it's the r⃗ in τ⃗ = r⃗ × F⃗. That double life is the whole point. In Unit 1, r⃗ is about translational motion, and calculus turns it into velocity and acceleration. In Unit 5, r⃗ points from the rotation axis to where a force is applied, and the cross product turns it into torque. If you're shaky on what r⃗ means, both kinematics problems and rotational dynamics problems get harder than they need to be. Position vectors also let you keep components separate, which is why projectile and 2D motion problems split cleanly into x and y equations.
Keep studying AP® Physics C: Mechanics Unit 1
Scalars and Vectors (Unit 1)
Position is the textbook example of a vector, while its magnitude |r⃗| (distance from the origin) is a scalar. Exam questions love asking you to tell these apart, like spotting that speed |v⃗| is a scalar even though it comes from a vector function.
Velocity and acceleration from r⃗(t) (Unit 1)
Velocity is literally the derivative of the position vector, component by component. Given r⃗(t) = (3t²) î + (4t³) ĵ, you differentiate each component to get v⃗, then take the magnitude for speed. This calculus chain (r⃗ → v⃗ → a⃗) is the spine of Physics C kinematics.
Cross product and torque (Unit 5)
Torque is τ⃗ = r⃗ × F⃗, where r⃗ is the position vector from the axis of rotation to the point where the force acts. The cross product means only the part of F⃗ perpendicular to r⃗ produces torque, which is why pushing along the line of r⃗ does nothing.
Lever arm / moment arm (Unit 5)
The lever arm is the shortcut version of the position vector in torque problems. Instead of computing a full cross product, you use r sin θ, the perpendicular distance from the axis to the force's line of action. Same physics, geometric shortcut.
Position vectors show up most often in multiple-choice stems that hand you r⃗(t) in component form and ask you to do something with it. The classic moves are differentiating to find the velocity vector at a specific time, taking |v⃗| = √(vₓ² + v_y²) for speed, computing the magnitude of r⃗ at a given t, or identifying which quantity in the setup is a scalar. Circular motion in disguise is a favorite, where r⃗(t) = R cos(ωt) î + R sin(ωt) ĵ describes a circle and you're expected to recognize it. On free-response questions, you won't usually be asked to define a position vector, but you'll use it constantly, both when deriving motion from a given r⃗(t) and when setting up τ⃗ = r⃗ × F⃗ in rotation problems. Know your derivatives cold and keep components separate until the last step.
The position vector r⃗ tells you where an object is relative to the origin at one instant. Displacement Δr⃗ = r⃗_f − r⃗_i tells you the change in position between two instants. Position depends on where you put the origin; displacement doesn't, because the origin choice cancels in the subtraction. On the exam, |r⃗| is distance from the origin, while |Δr⃗| is how far the object ended up from where it started.
The position vector r⃗ = x î + y ĵ + z k̂ points from the origin of your coordinate system to the object's location.
Differentiating r⃗(t) once gives the velocity vector, and differentiating again gives the acceleration vector, component by component.
The magnitude |r⃗| = √(x² + y² + z²) is a scalar, even though r⃗ itself is a vector.
In torque problems, r⃗ is the vector from the rotation axis to the point where the force is applied, and τ⃗ = r⃗ × F⃗.
A position vector like R cos(ωt) î + R sin(ωt) ĵ describes circular motion of radius R, and recognizing that pattern saves time on MCQs.
Position depends on your choice of origin, but displacement Δr⃗ = r⃗_f − r⃗_i does not.
It's the vector r⃗ from the origin of a coordinate system to a particle's location, written in components as x î + y ĵ + z k̂. In Physics C it usually appears as r⃗(t), a function of time you differentiate to get velocity and acceleration.
No. Position r⃗ locates an object relative to the origin at one moment, while displacement Δr⃗ = r⃗_f − r⃗_i is the change in position between two moments. Displacement doesn't depend on where you put the origin; position does.
Yes. |r⃗| = √(x² + y² + z²) is just a number (the distance from the origin), so it's a scalar. This exact distinction shows up in multiple-choice questions asking which quantity in a problem is a scalar versus a vector.
Differentiate each component of r⃗(t) with respect to time. For r⃗(t) = (3t²) î + (4t³) ĵ, velocity is v⃗ = (6t) î + (12t²) ĵ, and speed is the magnitude √((6t)² + (12t²)²).
Torque is defined as τ⃗ = r⃗ × F⃗, where r⃗ runs from the rotation axis to the point where the force acts. The cross product picks out only the perpendicular part of the force, which is why a longer r⃗ (a longer lever arm) gives more torque for the same force.
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