Triangle Law of Vector Addition

What is Vector Addition?

Triangle law of vector addition is one of the vector addition law. Vector addition is defined as the geometrical sum of two or more vectors as they do not follow regular laws of algebra. The resultant vector is known as the composition of a vector.

There are few conditions that are applicable for any vector addition and they are:

  • Scalar and vectors can never be added.
  • For any two scalars to be added, they must be of the same nature. Example mass should be added with mass and not with time.
  • For any two vectors to be added, they must be of the same nature. Example velocity should be added with velocity and not with force.

There are two laws of vector addition and they are:

  • Triangle law of vector addition
  • Parallelogram law of vector addition

What is Triangle Law of Vector Addition?

Triangle law of vector addition states that when two vectors are represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector.

Triangle law of vector addition formula

\(\underset{R}{\rightarrow}=\underset{A}{\rightarrow}+\underset{B}{\rightarrow}\)

Triangle Law of Vector Addition

To obtain \(\underset{R}{\rightarrow}\) which is the resultant of the sum of \(\underset{A}{\rightarrow}\) and \(\underset{B}{\rightarrow}\) with the same order of magnitude and direction as shown in the figure.

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Triangle Law of Vector Addition Derivation

Consider two vectors P and Q that are represented in the order of magnitude and direction by the sides OA and AB respectively of the triangle OAB. Let R be the resultant of vectors P and Q.

Derivation of triangle law of vector addition

\(R=P+Q\)

From triangle OCB,

\(OB^{2}=OC^{2}+BC^{2}\) \(OB^{2}=(OA+AC)^{2}+BC^{2}\) (eq.1)

In triangle ACB with ϴ as the angle between P and Q

\(cos\Theta =\frac{AC}{AB}\) \(AC=ABcos\Theta =Qcos\Theta\) \(sin\Theta =\frac{BC}{AB}\) \(BC=ABsin\Theta =Qsin\Theta\) \(R^{2}=(P+Qcos\Theta )^{2}+(Qsin\Theta )^{2}\) (after substituting AC and BC in eq.1)

\(R^{2}=P^{2}+2PQcos\Theta +Q^{2}cos^{2}\Theta +Q^{2}sin^{2}\Theta\) \(R^{2}=P^{2}+2PQcos\Theta +Q^{2}\)

therefore, \( R=\sqrt{P^{2}+2PQcos\Theta +Q^{2}}\)

Above equation is the magnitude of the resultant vector.

To determine the direction of the resultant vector, let ɸ be the angle between the resultant vector R and P.

From triangle OBC,

\(tan\phi =\frac{BC}{OC}=\frac{BC}{OA+AC}\) \(tan\phi =\frac{Qsin\Theta }{P+Qcos\Theta }\)

therefore, \( \phi =tan^{-1}(\frac{Qsin\Theta }{P+Qcos\Theta })\)

Above equation is the direction of the resultant vector.

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