Cartesian Vectors - Engineering Mechanics - Civil Engineering (CE)

Cartesian Vectors

Vectors are directed quantities that represent both magnitude and direction. In the Cartesian (rectangular) coordinate system a vector is represented by its components along three mutually perpendicular axes, usually labelled x, y and z. A Cartesian vector therefore provides how much to move along each axis to reach the head of the vector from its tail.

• For example, the vector ⟨3, 2, -1⟩ means a displacement of 3 units in the positive x-direction, 2 units in the positive y-direction and 1 unit in the negative z-direction.
• Cartesian vectors are used throughout engineering mechanics to describe displacements, velocities, accelerations and forces; they are particularly convenient because vector addition and resolution reduce to component-wise operations along the x, y and z axes.

Cartesian Vector

Cartesian Force Vectors

When a force acts on a body, it can be described as a Cartesian force vector whose components are the projections of that force on the coordinate axes. The Cartesian components of a force give the magnitude and sense of the force in each coordinate direction.

• A force given by the component triple ⟨Fx, Fy, Fz⟩ means the force has magnitude Fx along the positive x-axis, Fy along the positive y-axis and Fz along the positive z-axis (a negative component indicates the opposite sense).
• Example: the vector ⟨10, 5, -3⟩ represents a force of 10 units in +x, 5 units in +y and 3 units in -z.
• Any force F can be written in the Cartesian vector (component) form using the orthonormal unit vectors i, j and k as

F = F_xi + F_yj + F_zk

where i, j and k are the unit vectors in the x, y and z directions respectively.

Expressing a Force Vector Using Magnitude and Two Points on Its Line of Action

Often the force is specified by its magnitude and the line along which it acts; the line of action can be defined by two points through which it passes. Using position vectors for these points we obtain the Cartesian direction of the force.

• Let P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) be two points on the line of action.
• Let r₁ be the position vector of P and r₂ be the position vector of Q, so that the vector from P to Q is r = r₂ - r₁.
• Writing the position vectors in Cartesian form gives

r₁ = x₁i + y₁j + z₁k

r₂ = x₂i + y₂j + z₂k

Therefore

r = (x₂ - x₁)i + (y₂ - y₁)j + (z₂ - z₁)k       (2.2)

The components of r are obtained by subtracting the coordinates of the tail from those of the head. If the vector from Q to P is desired, the sign of r is reversed.

If the magnitude of the force is F and its line of action has a unit direction vector s, the force vector may be written as

F = Fs

The unit vector s along the line of action may be obtained by normalising r:

s = r / |r|

Alternatively, the direction of s can be given using the coordinate direction cosines (or direction cosines):

s = cosα i + cosβ j + cosγ k

and the direction cosines satisfy

cos²α + cos²β + cos²γ = 1

Addition of Forces

For a system of concurrent forces, the resultant is the vector sum of all forces. Component-wise addition is the standard and simplest method in Cartesian coordinates.

If P is the resultant of forces F₁, F₂, F₃, ... then

P = ΣF_i

Expanding the components gives

P = (F_1x + F_2x + ...)i + (F_1y + F_2y + ...)j + (F_1z + F_2z + ...)k

or compactly

P = ΣF_ixi + ΣF_iyj + ΣF_izk       (2.5)

Here ΣF_ix, ΣF_iy and ΣF_iz are the algebraic sums of the force components in the corresponding axes; signs must reflect direction.

Resolution of Forces

Resolution of a force means replacing it by two or three component forces whose vector sum equals the original force. The method chosen depends on the directions into which the force is to be resolved.

Parallelogram Law

A single force can be resolved into two component forces acting in any two given directions by constructing a parallelogram with the given force as the diagonal and the desired component directions as the sides. The components are the two non-parallel sides of that parallelogram.

• Resolution is the inverse of the process of vector addition-if two forces are known, their resultant is the diagonal of the parallelogram constructed on them.
• Six quantities are involved in the general resolution problem: the magnitudes and directions of the resultant and of the two components; if four are known the other two can be determined.

Rectangular Components (Perpendicular Resolution)

When a force is resolved into perpendicular directions (for example into x and y components in a plane), the components are called rectangular components.

For a planar force of magnitude F making an angle θ with the positive x-axis, the rectangular components are

F_x = F cosθ

F_y = F sinθ

If both components are known, the angle can be obtained from

θ = tan^-1(F_y / F_x)

These relations generalise to three dimensions when components along x, y and z axes are required.

Addition of Concurrent Forces Using Components

Concurrent forces (forces whose lines of action meet at a common point) can be added easily by summing their Cartesian components. For two concurrent forces F₁ and F₂ acting at the same point O, the resultant P is

P = F₁ + F₂

In components, this is

P_x = F_1x + F_2x

P_y = F_1y + F_2y

P_z = F_1z + F_2z

The magnitude and direction of P follow from these summed components.

Practical Notes and Applications

• Cartesian vector methods are standard in structural analysis, rigid-body mechanics, dynamics and statics; they simplify equilibrium equations by turning vector equations into three scalar equations (one for each axis).
• Care with signs and consistent choice of coordinate axes are essential; draw a clear free-body diagram and resolve each force into components before algebraic summation.
• Direction cosines provide a compact way to express 3D directions and to construct unit vectors when two points on a line are known.

Summary: Cartesian vectors describe directed quantities by their components along orthogonal axes. Forces are expressed as Cartesian vectors using components or by magnitude and a unit direction vector. Resolution and addition of forces reduce to component operations; the parallelogram law and rectangular component relations provide the geometric and algebraic bases for these operations.