Chapter 2 · Vectors CHAPTER 3 · PARTICLE EQUILIBRIUM →

Parallelogram Law

When two vectors act at a common point, their resultant may be determined graphically using the parallelogram law. This construction provides a clear geometric interpretation of vector addition and forms the basis for trigonometric solution methods used in statics.

Parallelogram Law — Statement

If two vectors act from the same point, their resultant is represented by the diagonal of the parallelogram formed by drawing lines parallel to each vector through the tip of the other.

Parallelogram Properties

These geometric properties ensure that vectors may be translated without altering their magnitude or direction.

Opposite sides are equal in length.
Opposite interior angles are equal.
The sum of the four interior angles is \(360^\circ\).

Graphical Construction of the Resultant

Procedure Geometric Construction

Place vectors A and B tail-to-tail.
Draw a line parallel to A through the tip of B.
Draw a line parallel to B through the tip of A.
The diagonal from the common tail represents the resultant R.

Step-by-step construction of the parallelogram law — Graphical construction of the resultant using the parallelogram law.

Trigonometric Solution of the Resultant

The triangle formed by the sides of the parallelogram may be analyzed using trigonometry to determine the magnitude and direction of the resultant vector.

Trigonometry Resultant Vectors

Magnitude — Law of Cosines

If two vector magnitudes and the included angle are known, the magnitude of the resultant may be determined using:

\[ C^2 = A^2 + B^2 - 2AB\cos\gamma \]

This relation is derived directly from the geometry of the parallelogram.

Direction — Law of Sines

Once the magnitude of the resultant is known, its direction relative to either vector may be obtained from:

\[ \frac{A}{\sin\alpha} = \frac{B}{\sin\beta} = \frac{C}{\sin\gamma} \]

Use \(\alpha\) or \(\beta\) to describe the direction of the resultant relative to vector \(A\) or \(B\).

Triangle used for law of sines and law of cosines — Triangle extracted from the parallelogram for trigonometric analysis.

ℹ️

Which law should be applied first?

Law of Cosines: Apply first when two sides and the included angle are known.
Law of Sines: Apply first only when one side and all angles are known.

In most statics problems, the law of cosines is applied first, followed by the law of sines to determine direction.

CAUTION. When the law of sines is applied in an SSA configuration, two mathematically valid solutions may exist. The physically correct solution must be identified from the vector geometry. To avoid ambiguity, apply the law of cosines first whenever possible.

Worked Example — Parallelogram Law

Example Parallelogram Law

Problem

Two forces act at a point: \(F_1 = 80\,\text{lb}\) and \(F_2 = 60\,\text{lb}\). The included angle between the forces is \(120^\circ\). Determine the magnitude of the resultant \(|R|\) and its direction measured from the \(+x\)-axis.

Use the parallelogram law to form the equivalent triangle. Apply the law of cosines to compute \(|R|\), then apply the law of sines to compute the direction.

Worked example: parallelogram law with law of sines and cosines — Graphical construction and trigonometric solution for \(|R|\) and direction.

Form the triangle from the parallelogram

The parallelogram construction produces a triangle with sides \(60\), \(80\), and resultant \(R\). The included angle in the triangle is \(60^\circ\).

Triangle extracted from parallelogram for cosine-law setup — The \(60^\circ\) angle follows from parallelogram geometry: opposite interior angles are equal, and the sum of the four interior angles is \(360^\circ\).

Magnitude using the Law of Cosines

\[ R = \sqrt{60^2 + 80^2 - 2(60)(80)\cos(60^\circ)} = 72.1 \text{lb} \]

Direction using the Law of Sines

Using the triangle,

Triangle labeled for law of sines showing alpha and R

\[ \frac{60}{\sin(\alpha)} = \frac{72.1}{\sin(60^\circ)} \] \[ \alpha = 46.1^\circ \]

CAUTION. Do not assume the angle of the resultant from how the diagram looks. Even if the resultant appears to be “in the middle,” its angle must be calculated from the geometry, not guessed from the picture.

Determine the direction relative to the horizontal

Once \(\alpha\) is known, the direction of the resultant measured from the horizontal is obtained from the geometry of the diagram:

\[ \theta = 120^\circ - \alpha = 73.9^\circ \]

Reference angles used to determine direction from the horizontal

Final Answers:

\(|R| = 72.1\,\text{lb}\)
\(\alpha = 73.9^\circ\) (from \(+x\))

Worked Example — Vertical Resultant

Example Parallelogram Law Vertical Resultant

Problem

A 600 lb force acts at an angle of 60° from the positive \(x\)-axis. A second force \(F\) acts at an unknown angle \(\theta\). The two forces combine to produce a resultant of 1200 lb acting vertically upward.

Determine the value of \(\theta\) and the magnitude of \(F\).

Parallelogram law example with one horizontal force

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