Vectors in Engineering Mechanics – Forces, Components & Vector Operations

Vectors provide a mathematical method for representing quantities that have both magnitude and direction. In engineering mechanics they are used extensively for displacement, velocity, acceleration, force, moment and many other physical quantities.

Need help choosing a mechanics or stress analysis method? See the Mechanics and Stress Analysis Guide for guidance on forces, motion, loading, stress, strain, yielding, buckling, pressure vessels, rotating components, torsion and plate analysis.

This page provides a reference to vector representation, coordinate systems, vector components, vector addition, scalar and vector products, vector identities and differentiation.

Vector Quantities and Forces

At the most basic level in mechanics we deal with quantities such as displacement, velocity, acceleration and force.

A vector quantity is fully described by both magnitude and direction. A scalar quantity is fully described by magnitude alone.

Vector quantities	Scalar quantities
Displacement	Distance
Velocity	Speed
Acceleration	Mass
Force	Time
Moment / torque	Energy

Vector quantities can be represented graphically by arrows. The length of the arrow represents the magnitude of the vector to a chosen scale, while the orientation of the arrow represents its direction.

Vectors are particularly important when representing forces acting on engineering components. A diagram showing the relevant forces and moments acting on an isolated body is known as a free-body diagram.

Coordinate Systems in Engineering

For one-dimensional problems, a single coordinate and a defined positive direction are normally sufficient.

For two-dimensional problems the most commonly used coordinate systems are:

Cartesian coordinates – normally defined by mutually perpendicular x and y axes.
Polar coordinates – defined by a radial coordinate r and an angular coordinate θ measured from a specified reference axis.

For three-dimensional engineering problems, common coordinate systems include:

Cartesian coordinates – defined by mutually perpendicular x, y and z axes.
Cylindrical coordinates – defined by radial coordinate r, angular coordinate θ and axial coordinate z.
Spherical coordinates – defined by a radial distance and two angular coordinates. The precise angular convention should always be stated because different disciplines use different definitions for the two angles.

Cartesian, cylindrical and spherical coordinate systems

Vector Representation

A vector has both magnitude and direction. A free vector may be moved parallel to itself without changing its meaning, whereas vectors associated with a particular point, line of action or position must retain the appropriate positional information.

Vectors may be manipulated graphically or algebraically. Graphical construction can be useful for visualising simple two-dimensional problems, but algebraic vector methods are generally preferred for accurate engineering calculations and become particularly useful in three-dimensional systems.

Graphical Addition and Resolution

Typical graphical methods for adding and resolving vectors are shown below.

Graphical vector addition and resolution using triangle and parallelogram methods

Two or more vectors can be added to form a single resultant. Conversely, a vector can be separated into components by resolving the vector along selected directions.

For example, a force acting at an angle can be resolved into horizontal and vertical components, allowing the effects in each coordinate direction to be analysed separately.

Algebraic Representation

A vector is conventionally represented by a bold symbol such as A. Its magnitude is written as |A| or simply A where the context is clear.

A unit vector has magnitude one and defines a direction. In Cartesian coordinates, the unit vectors in the x, y and z directions are conventionally written i, j and k.

A vector A can therefore be written in terms of its Cartesian components:

A = A_{x} i + A_{y} j + A_{z} k

The magnitude of the vector is:

|A| = \sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}}

Vector Addition

Two vectors are added by adding their corresponding components:

A + B = (A_{x} + B_{x}) i + (A_{y} + B_{y}) j + (A_{z} + B_{z}) k

Vector addition is commutative:

A + B = B + A

Multiplication by a Scalar

Multiplying a vector by a scalar multiplies each component by that scalar:

c A = c A_{x} i + c A_{y} j + c A_{z} k

For a positive scalar the vector retains its direction. For a negative scalar the vector direction is reversed. Multiplication by zero produces the zero vector.

Scalar or Dot Product

The scalar product, or dot product, of vectors A and B is:

A \cdot B = |A| |B| \cos θ

where θ is the angle between the vectors, with 0 ≤ θ ≤ π.

In Cartesian component form:

A \cdot B = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}

The result of a dot product is a scalar.

An important engineering example is mechanical work. For a constant force F acting through displacement s:

W = F \cdot s

Vector or Cross Product

The vector product, or cross product, of two vectors is:

A \times B = |A| |B| \sin θ n

where n is a unit vector perpendicular to the plane containing A and B. Its direction is determined by the right-hand rule.

Unlike the dot product, the result of a cross product is a vector. The order of the vectors is important:

A \times B = - (B \times A)

An important engineering application is the moment of a force about a point:

M = r \times F

where r is the position vector from the point about which the moment is taken to a point on the force's line of action.

Right-hand rule for the cross product of two vectors

For A × B, curl the fingers of the right hand from A towards B through the smaller angle. The thumb indicates the direction of the resulting cross-product vector.

Cross Product in Cartesian Coordinates

For Cartesian unit vectors:

i \times j = k

j \times k = i

k \times i = j

Reversing the order reverses the sign:

j \times i = - k, k \times j = - i, i \times k = - j

Also:

i \times i = j \times j = k \times k = 0

For:

A = A_{x} i + A_{y} j + A_{z} k

and

B = B_{x} i + B_{y} j + B_{z} k

the cross product is:

A \times B = (A_{y} B_{z} - A_{z} B_{y}) i + (A_{z} B_{x} - A_{x} B_{z}) j + (A_{x} B_{y} - A_{y} B_{x}) k

Determinant Form of the Cross Product

The same relationship can be written compactly as:

A \times B = | \begin{matrix} i & j & k \\ A_{x} & A_{y} & A_{z} \\ B_{x} & B_{y} & B_{z} \end{matrix} |

General Vector Identities

Addition

Vector addition is both commutative and associative:

A + B = B + A

(A + B) + C = A + (B + C)

Scalar Multiplication

a (A + B + C) = a A + a B + a C

Scalar Triple Product

The scalar triple product is invariant under cyclic permutation:

A \cdot (B \times C) = B \cdot (C \times A) = C \cdot (A \times B)

Vector Triple Products

Cross products are not associative, so the brackets are important.

(A \times B) \times C = B (A \cdot C) - A (B \cdot C)

and:

A \times (B \times C) = B (A \cdot C) - C (A \cdot B)

Differentiation of Vectors

A vector whose components vary with a parameter such as time can be differentiated component by component.

If:

A = A_{x} i + A_{y} j + A_{z} k

and the Cartesian unit vectors are fixed in direction, then:

\frac{d A}{d t} = \frac{d A_{x}}{d t} i + \frac{d A_{y}}{d t} j + \frac{d A_{z}}{d t} k

This is fundamental to engineering kinematics. For example, differentiating the position vector r with respect to time gives velocity:

v = \frac{d r}{d t}

and differentiating velocity gives acceleration:

a = \frac{d v}{d t} = \frac{d^{2} r}{d t^{2}}

Note: The simple component differentiation above assumes a fixed Cartesian coordinate system. In rotating or otherwise moving coordinate systems the unit vectors may themselves vary with time and their derivatives must also be included.

Related RoyMech Mechanics References

External Vector References

Vector Components and Addition – NASA – short introduction to vector components and addition.
Vectors – Wolfram MathWorld – detailed mathematical reference for vectors and vector operations.

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