Position Vectors - Engineering Mechanics - Civil Engineering (CE)

Introduction

The position vector of a point is a directed line segment that locates the point with respect to a chosen reference or origin. In mechanics, it describes the position of a particle or a point of a body relative to a fixed reference frame. As the point moves, the position vector changes in length, direction, or both.

Key qualitative properties

• A position vector has one end fixed at the chosen origin and the other end at the point whose position is described.
• Changes in the position vector occur only as a change in magnitude (length), a change in direction (rotation), or both.
• A change in length indicates a change in the distance from the origin; a change in direction indicates rotation about the origin.
• The velocity of the point is defined as the time rate of change of the position vector.
• For motion constrained to a straight path, it is convenient to choose the position vector along that line; the velocity then lies along the same line and equals the time rate of change of the magnitude of the position vector.
• For motion on a circle, it is convenient to choose the position vector along the radius; the instantaneous velocity is perpendicular to the radius and arises from the rate of change of the direction of the position vector.
• For general curved motion, the position vector changes in both magnitude and direction; the instantaneous velocity is the vector sum of the rate of change of magnitude (radial component) and the rate of change of direction (transverse component).

Mathematical representation

Choose a Cartesian origin O and orthonormal basis vectors i, j, k. The position vector of a point P is written as

r = x(t) i + y(t) j + z(t) k

where x(t), y(t), z(t) are the coordinates of P as functions of time.

Magnitude and unit vector

The magnitude (distance from origin) is

|r| = √(x(t)² + y(t)² + z(t)²)

A unit vector in the direction of r is

e_r = r / |r|

Velocity and acceleration (Cartesian form)

Differentiate the position vector with respect to time to obtain the velocity vector.

r(t) = x(t) i + y(t) j + z(t) k

v(t) = dr/dt = dx/dt i + dy/dt j + dz/dt k

Differentiate velocity to obtain acceleration.

a(t) = d₂r/dt₂ = d₂x/dt₂ i + d₂y/dt₂ j + d₂z/dt₂ k

Interpretation of velocity components

• The component of velocity in the direction of the position vector gives the rate of change of the distance from the origin (radial speed).
• The component of velocity perpendicular to the position vector gives the transverse (or tangential) speed due to rotation of the direction of r.
• Thus, instantaneous velocity in general curved motion is the vector sum of radial and transverse components.

Motion on a straight line

When motion is along a fixed straight line through the origin (or when the position vector is chosen along the path), r = s(t) e where e is a fixed unit vector along the line.

v = dr/dt = ds/dt e

Here ds/dt is the scalar speed along the line and v has the same direction as the line.

Circular motion (radius vector representation)

For motion on a circle of radius R about the origin, the magnitude |r| = R is constant and only the direction of r changes with time.

Let the angular coordinate be θ(t). Then

r = R e_r

Differentiate r with respect to time using the polar unit-vector relations given below to obtain the velocity perpendicular to r.

Polar (plane) coordinates and unit-vector derivatives

In plane motion use polar coordinates (r, θ) with unit vectors e_r (radial) and e_θ (transverse). The position vector is

r = r e_r

The time derivatives of the unit vectors are

d(e_r)/dt = θ̇ e_θ

d(e_θ)/dt = -θ̇ e_r

Differentiate the position vector to get velocity and acceleration.

v = dr/dt = ṙ e_r + r θ̇ e_θ

a = d₂r/dt₂ = (r̈ - r θ̇²) e_r + (r θ̈ + 2 ṙ θ̇) e_θ

Here ṙ = dr/dt, r̈ = d₂r/dt₂, θ̇ = dθ/dt, and θ̈ = d₂θ/dt₂.

Physical meaning of polar acceleration terms

• The term r̈ is the radial acceleration due to change in the magnitude of r.
• The term -r θ̇² is the centripetal acceleration directed toward the centre due to change in direction of r.
• The term r θ̈ is the transverse acceleration due to angular acceleration.
• The term 2 ṙ θ̇ is the Coriolis (transverse) acceleration arising when the radial distance changes while the point has angular velocity.

Position vector and rigid-body kinematics (brief)

For a rigid body, the position vector locates any point of the body relative to a chosen origin. When the body undergoes translation and rotation, the velocity of a point P with position vector r relative to a body-fixed origin O is expressed as the sum of the velocity of O and the rotational contribution.

v_P = v_O + ω × r

Here ω is the angular velocity vector of the rigid body and × denotes the vector (cross) product.

Worked example

Find the velocity of a particle whose position vector is r(t) = (3t) i + (4) j (units: metres, time in seconds).

r(t) = 3t i + 4 j

dr/dt = 3 i + 0 j

Therefore the velocity is

v = 3 i m/s

Applications in engineering mechanics

• Particle kinematics: position vectors provide instantaneous location, velocity and acceleration of particles moving in space or on constrained paths.
• Mechanism analysis: position vectors locate points on links; differentiation gives link velocities and accelerations required in mechanism synthesis and dynamics.
• Rigid-body dynamics: expressing velocities as v = v_O + ω × r is central to analysing rotating machinery and structures.
• Projectile and motion under forces: position vectors are the primary unknowns in solving equations of motion under given forces.

Summary

The position vector is the fundamental quantity that locates a point with respect to an origin. Differentiation of the position vector yields velocity and acceleration. Choosing an appropriate coordinate system (Cartesian, polar, or body-fixed) simplifies the description of motion for straight-line, circular, or general curved paths and is essential for solving practical engineering kinematics problems.