Triangle Law of Vector Addition - Examples, FAQs

What is Vector Addition?

Vector addition is the process of finding the geometrical sum of two or more vectors. Unlike regular numbers, vectors have both magnitude and direction, so they do not follow ordinary algebraic rules. The vector obtained by combining two or more vectors is called the resultant vector or the composition of vectors.

Important Conditions:

• Scalars and vectors cannot be added.
• Only vectors of the same kind can be added (e.g., velocity with velocity, force with force).

Laws of Vector Addition:

• Triangle Law of Vector Addition
• Parallelogram Law of Vector Addition

Triangle Law of Vector Addition

The triangle law of vector addition states that if two vectors are represented by the sides of a triangle taken in order of magnitude and direction, then the resultant sum of the vectors is given by the triangle's third side in reverse order of magnitude and direction.

Mathematically, if you have two vectors A and B, and you place the tail of vector B at the head of vector A, then the resultant vector R is the vector drawn from the tail of A to the head of B. The direction and magnitude of this resultant vector represent the combined effect of both vectors.

Derivation of the Triangle Law of Vector Addition

Consider two vectors P and Q, represented in both magnitude and direction by the sides OA and AB of a triangle OAB, respectively. Let R be the product/resultant of the triangle law of vector additions. The resultant of P and Q is therefore represented by side OB according to the triangle law of addition or triangle law of vector addition.

We have,

$
R=P+Q
$

Expand A to C and draw perpendicular BC .
From triangle OCB we have,

$
\begin{aligned}
& \mathrm{OB}^2=\mathrm{OC}^2+\mathrm{BC}^2 \\
& \mathrm{OB}^2=(\mathrm{OA}+\mathrm{OC})^2+\mathrm{BC}^2
\end{aligned}
$
In the triangle ACB,

$
\begin{aligned}
& \cos \theta=A C / A B \\
& A C=A B \cos \theta=Q \cos \theta
\end{aligned}
$
Also,

$
\sin \theta=B C / A B
$
$
B C=A B \sin \theta=Q \sin \theta
$

Resultant magnitude:
Substitute the values for $A C$ and $B C$ in (i),

$
\begin{aligned}
& R^2=(P+Q \cos \theta)^2+(Q \sin \theta)^2 \\
& R^2=P^2+2 P Q \cos \theta+Q^2 \\
& R=\sqrt{ }\left(P^2+2 P Q \cos \theta+Q^2\right)
\end{aligned}
$
The above equation gives us the magnitude of the resultant.

Parallelogram Law of Vector Addition

The Parallelogram Law of Vector Addition is another fundamental method for determining the resultant of two vectors. This law states that if two vectors are represented as adjacent sides of a parallelogram, then the diagonal of the parallelogram that starts from the common point of the vectors represents their resultant.

Here's how it works:

1. Draw the Vectors: Start by placing the vectors A and B such that they both originate from the same point.
2. Form a Parallelogram: Complete the parallelogram by drawing lines parallel to A and B from the endpoints of each vector.
3. Draw the Resultant: The diagonal of the parallelogram that extends from the common point of the vectors to the opposite corner represents the resultant vector R.

Triangle Law of Forces

The Triangle Law of forces is applicable when there are three forces acting on a body in equilibrium. The two forces are then represented as two sides of a triangle in the same order, with their magnitude scaled to a suitable scale, and the resultant in the opposite order is the third side or closing side of the triangle.

The Triangle Law of Forces can also be used to calculate the resultant of two forces acting at a point.

What is the application of triangle law of vector addition?

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Summary

The sum of two vectors can be found mathematically using the triangular law of vector addition. Mathematical physics includes vector addition and subtraction as essential concepts. A vector is an entity with both magnitude and direction, sometimes known as a quantity. It is a geometric addition of the vectors.