The parallelogram law of vector addition is a fundamental principle for finding the combined effect (resultant) of two vectors acting at the same point.
Understanding the Parallelogram Law
According to the parallelogram law:
If two vectors are acting simultaneously at a point, then it can be represented both in magnitude and direction by the adjacent sides drawn from a point.
Consider two vectors, vector A and vector B, acting at a point. We can represent these vectors as adjacent sides of a parallelogram starting from that point.
Therefore, the resultant vector is completely represented both in direction and magnitude by the diagonal of the parallelogram passing through the point.
This diagonal, starting from the point where the two vectors originate, represents the single vector that has the same effect as the combination of vectors A and B.
Finding the Magnitude of the Resultant Vector
Let the two vectors be A and B, and let the angle between them be $\theta$. If R is the resultant vector obtained by adding A and B using the parallelogram law, its magnitude, denoted by |R| or R, can be found using the law of cosines.
Consider the parallelogram formed by vectors A and B. The diagonal representing the resultant R is the third side of a triangle formed by vector A, vector B (shifted parallel to itself), and the resultant R. The angle opposite to the resultant R in this triangle is $180^\circ - \theta$.
Using the law of cosines:
$R^2 = A^2 + B^2 - 2AB \cos(180^\circ - \theta)$
Since $\cos(180^\circ - \theta) = -\cos(\theta)$, the formula becomes:
$R^2 = A^2 + B^2 - 2AB (-\cos \theta)$
$R^2 = A^2 + B^2 + 2AB \cos \theta$
Taking the square root of both sides gives the magnitude of the resultant vector:
$$|\mathbf{R}| = \sqrt{A^2 + B^2 + 2AB \cos \theta}$$
Where:
- $A$ is the magnitude of vector A.
- $B$ is the magnitude of vector B.
- $\theta$ is the angle between vectors A and B.
Determining the Direction of the Resultant Vector
The direction of the resultant vector R is typically described by the angle it makes with one of the original vectors, say vector A. Let this angle be $\alpha$.
Consider the same triangle used for magnitude calculation. We can use the law of sines to find the angle $\alpha$.
$$\frac{|\mathbf{R}|}{\sin(180^\circ - \theta)} = \frac{B}{\sin \alpha}$$
Since $\sin(180^\circ - \theta) = \sin(\theta)$, we have:
$$\frac{R}{\sin \theta} = \frac{B}{\sin \alpha}$$
Rearranging this equation to solve for $\sin \alpha$:
$$\sin \alpha = \frac{B \sin \theta}{R}$$
Alternatively, we can directly write the formula for the tangent of the angle $\alpha$ that R makes with A:
$$\tan \alpha = \frac{B \sin \theta}{A + B \cos \theta}$$
This formula directly gives the tangent of the angle. To find the angle $\alpha$, you would take the arctangent (inverse tangent):
$$\alpha = \tan^{-1} \left( \frac{B \sin \theta}{A + B \cos \theta} \right)$$
Similarly, if you want the angle $\beta$ that R makes with vector B, the formula would be:
$$\beta = \tan^{-1} \left( \frac{A \sin \theta}{B + A \cos \theta} \right)$$
Summary Table
Here's a quick summary:
| Aspect | Description | Formula |
|---|---|---|
| Law Concept | Vectors are adjacent sides, resultant is the diagonal from their origin point. (Based on reference) | Geometric representation |
| Magnitude ($R$) | The length of the resultant diagonal. | $R = \sqrt{A^2 + B^2 + 2AB \cos \theta}$ |
| Direction ($\alpha$) | The angle the resultant makes with vector A (or another reference). | $\alpha = \tan^{-1} \left( \frac{B \sin \theta}{A + B \cos \theta} \right)$ |
| Direction ($\beta$) | The angle the resultant makes with vector B (or another reference). | $\beta = \tan^{-1} \left( \frac{A \sin \theta}{B + A \cos \theta} \right)$ |
The parallelogram law provides a clear geometric method for adding two vectors, and the derived formulas allow for precise calculation of the magnitude and direction of the resulting vector.
