Vector Algebra

Vectorial Representation

Forces have an intensity (magnitude), a direction (line of action) and a verse (sense), and a single number is not sufficient to represent these aspects, so they need a special mathematical object to represent them: this object is known as vector.

Vector forces have a Cartesian representation: \[\mathbf{F} = F_x \, \mathbf{i} + F_y \, \mathbf{j} \] where

• \(\mathbf{F}\) is the vector object
• \(F_x, \, F_y\) are the force components in the \(x\) and \(y\) direction respectively
• \(\mathbf{i}\) and \(\mathbf{j}\) are the unit vectors in the \(x\) and \(y\) direction respectively

or a polar representation: \[\mathbf{F} = \left(F,\theta \right) = F \,e^{i\varphi} \] where

• \(F\) is the magnitude (intensity) of the force
• \(\theta\) is the angle in degrees indicating the direction of the force (with respect to a reference 0° axis).
• \(\varphi\) is the same angle in radians used for the exponential notation
• \( i=\sqrt{-1} \) is the imaginary unit.

Graphically, a vector can be represented with an arrow. For example, the force \(\mathbf{F}=3\mathbf{i}+4\mathbf{j}\) is the vector represented below (click the "Toggle Cartesian" button to verify). You can read this as "3 units in the x-direction (horizontal) and 4 units in the y-direction (vertical)".

The polar representation of the same vector force is \(\mathbf{F}=\left(5,53°\right) = 5\, e^{i 0.93}\) (click the "Toggle Polar" button to verify) and can be read as "magnitude 5 units at an angle 53 degrees from the horizontal".

Drag the arrowhead directly to change the vector and see how its Cartesian and polar representations update.

You can also convert between representations using basic trigonometry and Pythagoras' Theorem:

\[ \begin{array}{ll} \\ F=\sqrt{F_x^2+F_y^2} & \qquad & \qquad & F_x=F\cos(\varphi) \\ \varphi=\tan^{-1}\left( \frac{F_y}{F_x}\right) & \qquad & \qquad & F_y=F\sin(\varphi) \\ \end{array} \]

The left column converts Cartesian to Polar; the right converts Polar to Cartesian.

Scalar Multiplication (Scaling)

It is easy to see, using Cartesian representation, that the scale factor affects both components in the same way \[\mathbf{R} = \alpha\,\mathbf{F}=\left(\alpha F_x\right) \, \mathbf{i} + \left(\alpha F_y\right) \, \mathbf{j}\]

And as a result, it follows that the magnitude (length) is also scaled by the same amount \[\|\mathbf{F}\|=F=\sqrt{F_x^2+F_y^2}\] \[\|\mathbf{R}\|=R=\sqrt{\alpha^2F_x^2+\alpha^2F_y^2}=|\alpha|\sqrt{F_x^2+F_y^2}=|\alpha| F\]

Feel free to play with the applet above and note the following facts:

• if \(\alpha>1\), the resultant vector \(\mathbf{R}\) will be longer than the original \(\mathbf{F}\)
• if \(0 < \alpha < 1\) the resultant vector \(\mathbf{R}\) will be shorter than the original \(\mathbf{F}\)
• if \(\alpha=0\), the resultant vector \(\mathbf{R}\) is null (zero length, no magnitude)
• if \(\alpha < 0\), the resultant vector \(\mathbf{R}\) will be opposite the original \(\mathbf{F}\) (i.e. their angles are 180° apart)

Vector Sum

Vectors can be easily added together using their Cartesian representation: \[ \mathbf{A}+\mathbf{B}=\left(A_x\,\mathbf{i}+A_y\,\mathbf{j}\right)+\left(B_x\,\mathbf{i}+B_y\, \mathbf{j} \right) = \left(A_x+B_x\right)\,\mathbf{i}+\left(A_y+B_y\right)\,\mathbf{j} \] Graphically, you can resort to the parallelogram rule that states that the resultant vector is always the diagonal of the parallelogram formed by the two addends.

Another way of graphically sum multiple vectors is the so-called triangle rule, that states that the resultant vector is a vector connecting the start and end points of the polygon formed by all addends, placed consecutively.

Engineering Applications

Force, velocity, acceleration — any quantity with both magnitude and direction is a vector. Here are some areas where vector algebra is central:

• Structural Analysis: Forces in trusses, cables, and beams are represented as vectors. Equilibrium requires vector sums to equal zero.
• Navigation: GPS systems use vector addition to determine positions and velocities. Aircraft and marine navigation rely on vector calculations for course plotting.
• Electromagnetism: Electric and magnetic fields are vector quantities. Force calculations on charged particles require vector operations.
• Robotics: Robot arm positions and velocities are described using vectors. Path planning involves vector transformations and additions.
• Computer Graphics: 3D rendering, lighting calculations, and camera movements all use extensive vector algebra.
• Fluid Mechanics: Velocity fields in fluids are vector fields. Flow analysis requires vector differentiation and integration.

Summary of Key Equations

Practice Problems

Knowledge Check Quiz