JEE Main 2025 - Triangle Law of Vector Addition Definition, Derivation, Formula and Examples
The Triangle Law of Vector Addition is a fundamental concept in Mathematics and Physics that helps in determining the resultant of two vectors. This method involves arranging two vectors in sequence to form a triangle, with the third side representing the resultant vector. Widely used in mechanics, engineering, and everyday problem-solving, the Triangle Law simplifies the process of vector analysis and provides a clear visual representation of vector addition.
What is Vector Addition?
Vector Addition refers to the process of combining two or more vectors to produce a resultant vector. If two vectors are represented as $\vec{A}$ and $\vec{B}$, their sum $\vec{R}$ is given by:
$\vec{R} = \vec{A} + \vec{B}$
Where
$\vec{R}$ is the Resultant Vector of $\vec{A}$ and $\vec{B}$.
This operation follows certain rules, such as the
Commutative property $(\vec{A} + \vec{B} = \vec{B} + \vec{A})$ and the
Associative property $(\vec{A} + \vec{B}) + \vec{C}= \vec{A} + (\vec{B} + \vec{C})$.
Vector addition can be performed geometrically (using the head-to-tail method or parallelogram method) or algebraically by adding corresponding components of the vectors. It is widely used in linear algebra, geometry, and applied mathematics.
What is The Triangle Law of Vector Addition?
The Triangle Law of Vector Addition is a method used to find the resultant of two vectors. It states that if two vectors are represented as two sides of a triangle taken in sequence, their resultant is represented by the third side of the triangle taken in the opposite direction.
Steps to Apply The Triangle Law of Vector Addition:
Place the tail of the second vector at the head of the first vector.
Draw the third side of the triangle from the tail of the first vector to the head of the second vector.
This third side represents the resultant vector in both magnitude and direction.
Triangle Law of Vector Addition Diagram
Mathematically, if $\vec{A}$ and $\vec{B}$ are two vectors, their resultant $\vec{R}$ is given by:
$\vec{R} = \vec{A} + \vec{B}$
Triangle Law of Vector Addition Formula
The Triangle Law of Vector Addition formula helps calculate the resultant vector when two vectors are added. If two vectors$\vec{A}$ and $\vec{B}$ are represented as two sides of a triangle, the resultant vector $\vec{R}$ is given by:
$|vec{R}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}|\cos\theta}$
Where:
$\vec{R}|$: Magnitude of the resultant vector
$\vec{A}|$ and$\vec{B}|$: Magnitudes of the two vectors
$\theta$: Angle between the two vectors
Direction of the Resultant Vector:
The direction of $\vec{R}$, measured as an angle α\alphaα from $\vec{A}$, can be calculated using:
$\alpha = \tan^{-1}\left(\frac{|\vec{B}|\sin\theta}{|\vec{A}| + |\vec{B}|\cos\theta}\right)$
Triangle Law of Vector Addition Derivation
Statement: The Triangle Law of Vector Addition states that if two vectors are represented as two sides of a triangle taken in sequence, their resultant is given by the third side of the triangle taken in the reverse direction.
Triangle Law of Vector Addition Diagram
From the above figure,
P = Vector P
Q = Vector Q
OA = Magnitude of vector P
AB = Magnitude of vector Q
R = Sum of vector P and Q using triangle law of vector addition
$\theta=$ The angle between vectors P and Q
Extending the side of OA till point C. So that line BC is perpendicular to OC. The direction of the resultant vector R is given by the angle $\phi$.
From the right-angled triangle OBC,
$O B^{2}=O C^{2}+B C^{2}$
$O B^{2}=(O A+A C)^{2}+B C^{2}$ …(1)
From the right triangle ABC,
$\begin{align} \cos \theta &=\dfrac{A C}{A B} ; \sin \theta=\frac{B C}{A B} \\ A C &=A B \cos \theta \\ A C &=Q \cos \theta \\ B C &=A B \sin \theta \\ B C &=Q \sin \theta \\ A C &=Q \cos \theta \quad ; \quad B C=Q \sin \theta\dots(2) \end{align}$
Substituting values from equation(2) in equation (1). We get,
$\begin{align} R^{2} &=(P+Q \cos \theta)^{2}+(Q \sin \theta)^{2} \\ R^{2} &=P^{2}+Q^{2} \cos ^{2} \theta+2 P Q \cos \theta+Q^{2} \sin ^{2} \theta \\ R^{2} &=P^{2}+2 P Q \cos \theta+Q^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right) \end{align}$
We already knew that, $\sin ^{2} \theta+\cos ^{2} \theta=1$
$\begin{align} R^{2} &=P^{2}+2 P Q \cos \theta+Q^{2} \\ R &=\sqrt{P^{2}+2 P Q \cos \theta+Q^{2}} \end{align}$
The magnitude of the resultant vector R is given by the equation,
$R=\sqrt{P^{2}+2 P Q \cos \theta+Q^{2}}$
To find the direction of R, we are taking the right triangle OBC,
$\tan \phi=\dfrac{B C}{A C}$
From equation (2),
$\begin{align} \tan \phi &=\dfrac{Q \sin \theta}{(O A+O C)} \\ \tan \phi &=\dfrac{Q \sin \theta}{(P+Q \cos \theta)} \\ \phi &=\tan ^{-1}\left(\dfrac{Q \sin \theta}{(P+Q \cos \theta)}\right) \end{align}$
The direction of the resultant vector of A and B, which in our case is R is given as,
$\phi=\tan ^{-1}\left(\dfrac{Q \sin \theta}{P+Q \cos \theta}\right)$
Notes on Triangle Law of Vector Addition
When the magnitude and direction of two vectors can be represented by the two sides of a triangle in the same order, the resultant is represented by the third side of the triangle in the opposite order.
The magnitude of the resultant vector is given by,
$R=\sqrt{P^{2}+2 P Q \cos \theta+Q^{2}}$
The direction of the resultant vector R is given by,
$\phi=\tan ^{-1}\left(\dfrac{Q \sin \theta}{P+Q \cos \theta}\right)$
Where,
R = Resultant vector
P = Vector P
Q = Vector Q
$\theta=$ The angle between vectors P and Q
$\phi =$ The direction of the resultant vector R
Triangle Law of Vector Addition Examples
Example 1: Two vectors having magnitudes of 4 units and 5 units are to be added. The vectors mentioned in this question make an angle of 60° with each other. Find the magnitude and the direction of the resultant vector using the triangle law of vector addition?
Solution: The triangle law of vector addition formula is given as,
$R=\sqrt{P^{2}+2 P Q \cos \theta+Q^{2}}$
Let P=4, and Q=5.
Given that $\theta=60^{\circ}$, so using this we get,
$\begin{align}&R=\sqrt{4^{2}+5^{2}+2 \times 4 \times 5 \times \cos 60^{\circ}} \\ &\Rightarrow R=\sqrt{16+25+40 \times 0.5} \\ &\Rightarrow R=\sqrt{61} \\ &\Rightarrow R=7.81 \end{align}$
The direction is given as,
$\begin{align} &\phi=\tan ^{-1}\left(\dfrac{Q \sin \theta}{P+Q \cos \theta}\right) \\ &\Rightarrow \phi=\tan ^{-1}\left(\dfrac{5 \sin 60^{\circ}}{4+5 \cos 60^{\circ}}\right) \\ &\Rightarrow \phi \simeq \tan ^{-1}\left(\dfrac{4.33}{6.5}\right) \\ &\Rightarrow \phi \simeq 33.66^{\circ} \end{align}$
So the resultant of the given vectors is 7.81 units and the direction is approximately 33.66°.
Example 2: The magnitude of the resultant of two vectors having an angle of 60° between them is 8 units. One of the vectors has a magnitude of 2 units and the direction of the resultant is 45°. Find the magnitude of the second vector?
Solution: Let, P=2.
Given that $\varphi=45^{\circ}$ and $\theta=60^{\circ}$.
We have,
$\phi=\tan ^{-1}\left(\dfrac{Q \sin \theta}{P+Q \cos \theta}\right)$
Putting the values that we have, we can find the value of Q.
$\begin{align} &\tan 45^{\circ}=\dfrac{0.866 Q}{2+0.5 Q} \\ &\Rightarrow 1 \times(2+0.5 Q)=0.866 Q \\ &\Rightarrow 2=Q(0.866-0.5) \\ &\Rightarrow \dfrac{2}{0.366}=Q \\ &\Rightarrow Q \simeq 5.46 \end{align}$
So, the second vector is approximately equal to 5.46 units.
Conclusion
The triangle law of vector addition is a mathematical concept that is used to find the sum of two vectors. Vector addition and subtraction are integral parts of mathematical physics. A vector is a quantity, or it is also called an object that has both a magnitude and a direction. But a scalar is a quantity that has only magnitude and no direction. The process of adding two or more vectors is known as vector addition. The vectors are added geometrically. Triangle Law, Parallelogram Law and Polygon Law are the three laws for vector addition.
The triangle law for vector addition states that if two vectors are represented by two sides of a triangle taken in order, then their vector sum is represented by the third side of the triangle taken in the opposite direction.
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FAQs on Triangle Law of Vector Addition for JEE Main 2025
1. What are the other laws of vector addition other than the triangle law of vector addition?
The parallelogram law states that, If two vectors are represented by the two adjacent sides of a parallelogram, then the resultant vector is given by the diagonal of the parallelogram passing through the point of intersection of the two vectors. The polygon law of vector addition states that if the number of zero vectors is represented by the (n – 1) sides of an n-sided polygon, then the resultant vector is given by the closing side or the nth side of the polygon taken in the opposite order.
$R=\sqrt{P^{2}+2 P Q \cos \theta+Q^{2}}$
Where,
R = Resultant vector
P = Vector P
Q = Vector Q
$\theta =$ The angle between vectors P and Q
2. What is the Triangle Law of Forces?
The Triangle Law of Forces states that if two concurrent forces are acting simultaneously on a body and are represented in magnitude and direction by the sides of a triangle taken in order, then the third side of the triangle represents the resultant of the forces in magnitude and direction taken in the opposite order. This is essentially just the triangle law of vector addition, but it is applied to forces as forces are also vectors.
3. What is the Triangle Law Of Vector Addition?
The Triangle Law of Vector Addition states that if two vectors are represented as two sides of a triangle taken in sequence, their resultant is represented by the third side of the triangle taken in the opposite direction.
4. How do you apply the Triangle Law of Vector Addition?
To apply the Triangle Law, place the tail of the second vector at the head of the first vector. The resultant vector is represented by the side connecting the tail of the first vector to the head of the second vector.
5. What is the Triangle Law Of Vector Addition Formula?
Triangle Law Of Vector Addition Formula:
$\vec{R}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}|\cos\theta}$
Where $\theta$ is the angle between the two vectors.
6. What is the direction of the resultant vector in the Triangle Law?
The direction of the resultant vector is given by:
$\alpha = \tan^{-1}\left(\frac{|\vec{B}|\sin\theta}{|\vec{A}| + |\vec{B}|\cos\theta}\right)$
where $\alpha$ is the angle of the resultant concerning the first vector.
7. Can the Triangle Law of Vector Addition be used for three or more vectors?
No, the Triangle Law is specifically for two vectors. For three or more vectors, methods like the polygon law or successive addition must be used.
8. Is the Triangle Law of Vector Addition commutative?
Yes, vector addition follows the commutative property, i.e., $\vec{A} + \vec{B} = \vec{B} + \vec{A}$.
9. What is the difference between the Triangle Law and Parallelogram Law of Vector Addition?
The Triangle Law represents two vectors as two sides of a triangle, while the Parallelogram Law represents them as adjacent sides of a parallelogram. Both laws give the same resultant vector.
10. What is the significance of the angle in the Triangle Law of Vector Addition?
The angle $\theta$ between the vectors determines the magnitude and direction of the resultant. A smaller angle results in a larger resultant, while a larger angle reduces the resultant magnitude.
11. Why is there a negative sign in the Triangle Law of Vector Addition formula?
The negative sign represents the directional nature of vectors, but it is typically omitted when calculating the magnitude.
12. Where is the Triangle Law of Vector Addition used in real life?
The Triangle Law is used in physics and engineering to resolve forces, displacements, and velocities. It is also used in navigation, where different directional movements are combined to find the resultant path.
13. What happens when the angle between two vectors is $90^\circ$?
When$\theta = 90^\circ$, the resultant magnitude simplifies to:
$\vec{R}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2}$
as c$\cos 90^\circ = 0$.
14. Can the Triangle Law of Vector Addition be represented graphically?
Yes, the Triangle Law is often represented graphically by connecting vectors in sequence and completing the triangle to find the resultant vector.
15. How does the Triangle Law help in vector analysis?
The Triangle Law simplifies vector addition by providing a clear geometric representation and formula, making it easier to calculate the resultant vector.
