In the parallelogram of vectors method
Scalars & Vectors - OCR A-Level Physics
Key Definition
Parallelogram of vectors
A method for adding two vectors that act at the same point: draw both from the common point, complete the parallelogram using them as adjacent sides, and the diagonal from the common point is the resultant.
A method for adding two vectors that act at the same point: draw both from the common point, complete the parallelogram using them as adjacent sides, and the diagonal from the common point is the resultant.
Diagram pending
Parallelogram of forces. Two solid arrows $\mathbf{A}$ and $\mathbf{B}$ drawn from a common point $O$ at an angle $\theta$ between them. Dashed parallel lines complete the parallelogram. Diagonal $\mathbf{R}$ drawn from $O$ to the opposite corner, labelled as the resultant.
Will be replaced with a GeoGebra SVG in stream 2.
- In the parallelogram of vectorsA method for adding two vectors: draw them from the same point and complete the parallelogram; the diagonal is the resultant. method, both vectors are drawn to scale from the same point, with the correct angle between them.
- Complete the parallelogram by drawing two extra lines, each parallel to one of the original vectors and starting at the nose of the other.
- The resultantThe single vector that has the same effect as two or more vectors acting together. is the diagonal of the parallelogram drawn from the common origin to the opposite corner.
- This method is equivalent to the triangle method. Both give the same resultant; choose whichever is clearer for the problem.
- The parallelogram method is especially useful when two forces act at the same point on an object (e.g. two ropes pulling a barge).
$$R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$$
where $\theta$ is the angle between vectors $\mathbf{A}$ and $\mathbf{B}$ when drawn from the same point. Note the plus sign (the parallelogram form), in contrast to the minus sign in the triangle (cosine-rule) form, because the interior angle of the triangle is $180^\circ - \theta$.
- For two vectors at right angles ($\theta = 90^\circ$), the parallelogram is a rectangle. $\cos 90^\circ = 0$, so $R = \sqrt{A^2 + B^2}$ (Pythagoras).
- For two vectors along the same line ($\theta = 0^\circ$), $\cos 0^\circ = 1$, so $R = \sqrt{A^2 + B^2 + 2AB} = A + B$ (arithmetic addition).
- For two equal vectors ($A = B$), the diagonal bisects the angle between them, so the resultant points along the angle bisector.
Common Mistake
MEDIUM
Wrong: Drawing the diagonal between the two noses of the vectors (the "wrong" diagonal). This gives the vector $\mathbf{B} - \mathbf{A}$, not $\mathbf{A} + \mathbf{B}$.
Right: The resultant is the diagonal from the common origin $O$ to the opposite corner of the parallelogram. The other diagonal represents vector subtraction.
Right: The resultant is the diagonal from the common origin $O$ to the opposite corner of the parallelogram. The other diagonal represents vector subtraction.
Examiner Tips and Tricks
- If the question gives forces "acting at the same point", parallelogram is usually the cleanest method.
- If the question gives a sequence of displacements (a walks east, then north), the triangle (nose-to-tail) method is more natural.
- If asked to find the angle the resultant makes, finish the calculation by quoting the angle relative to a stated direction (e.g. "$23^\circ$ above the horizontal", not just "$23^\circ$").