Vector addition and Vector Subtraction - Overview, Structure, Properties & Uses

Vector Addition And Vector Subtraction

Edited By Vishal kumar | Updated on Jul 02, 2025 05:35 PM IST

Vectors are fundamental elements in physics and mathematics, representing quantities that have both magnitude and direction. Unlike scalar quantities, which can be simply added or subtracted using basic arithmetic, vector operations require special consideration of their directional components. Understanding vector addition and subtraction is essential for accurately describing physical phenomena such as forces, velocities, and displacements.

This article delves into the methods for combining and resolving vectors, starting with the straightforward cases of vectors in the same direction and progressing to more complex scenarios involving different directions, employing the triangle and parallelogram laws of vector addition.

Vector Addition

Vector quantities are not added according to simple algebraic rules, because their direction matters.
The addition of vectors means determining their resultant.
When two vectors are in the same direction then upon addition the direction of the resultant vector is the same as any of the two vectors, while the magnitude of the resultant vector is simply the algebraic sum of two vectors.
eg, Vector $\vec{A}$ has magnitude $4 \&$ vector $\vec{B}$has magnitude 2 in the same direction.

$ \vec{A}+\vec{B}=4+2=6 $

So resultant has a magnitude equal to 6 while its direction is either along $\vec{A}$ or $\vec{B}$

Vector Subtraction

Vector subtraction of $\vec{B}$ from $\vec{A}$ is equal to Vector addition of $\vec{A}$ and negative vector of $\vec{B}$.
$
\vec{A}-\vec{B}=\vec{A}+(-\vec{B})
$
E.g., Vector $\vec{A}$ and $\vec{B}$ are in the east direction with magnitudes 4 and 2 respectively.

Vector subtraction of $\vec{B}$ from $\vec{A}$ is equal
$
=\vec{A}-\vec{B}=4-2=2
$

The resultant vector has a magnitude of 2 in the east direction.

Now, let’s understand the different scenarios when both vectors do not have the same direction then the triangle law of vector addition and the Parallelogram law of vector addition come into play.

Triangle law of Vector Addition

If two vectors are represented by both magnitude and direction by two sides of a triangle taken in the same order then their resultant is represented by side of the triangle.

The figure represents the triangle law of vector addition So, the resultant side C is given by

$
c=\sqrt{a^2+b^2+2 a b \cos \theta}
$

Where $\theta=$ angle between two vectors.

Parallelogram law of Vector Addition

If two vectors are represented by both magnitude and direction by two adjacent sides of a parallelogram taken from the same point then their resultant is also represented by both magnitude and direction taken from the same point but by the diagonal of the parallelogram.

The figure represents the law of parallelogram Vector Addition

Commutative law

The Sum of vectors remains the same in whatever order they may be added.

$\vec{P}+\vec{Q}=\vec{Q}+\vec{P}$

Fig shows the Commutative law of vector addition.

Solved Example Based On Vector Addition And Vector Subtraction

For More Information On Vector Addition And Vector Subtraction, Watch The Below Video:

Now let’s understand the above law by some solved examples.

Example 1: The resultant of two forces 3P and 2P is R. If the first force is doubled, then the resultant is also doubled. The angle between the forces is

1) $120^{\circ}$
2) $60^{\circ}$
3) $180^{\circ}$
4) $90^{\circ}$

Solution:

From the triangle law of vector addition

So using the concept,

$
\begin{aligned}
& R^2=(3 P)^2+(2 P)^2+2 \cdot(3 P) \cdot(2 P) \cdot \cos \theta \\
& R^2=13 P^2+12 P^2 \cos \theta ....... (1)
\end{aligned}
$

When the first force is doubled, the resultant is doubled
$
\begin{aligned}
& \text { So, }(2 R)^2=(6 P)^2+(2 P)^2+2 \cdot(6 P)(2 P) \cdot \cos \theta \\
& \Rightarrow 4 R^2=36 P^2+4 P^2+24 P^2 \cos \theta \\
& \Rightarrow R^2=10 P^2+6 P^2 \cos \theta ............(2)
\end{aligned}
$

Equating (1) and (2)
$
\begin{aligned}
& \Rightarrow 13 P^2+12 P^2 \cos \theta=10 P^2+6 P^2 \cos \theta \\
& \Rightarrow 3 P^2=-6 P^2 \cos \theta \\
& \Rightarrow \cos \theta=-\frac{1}{2} \text { or } \theta=120^{\circ}
\end{aligned}
$

Equating (1) and (2)

$\begin{aligned}
& \Rightarrow 13 P^2+12 P^2 \cos \theta=10 P^2+6 P^2 \cos \theta \\
& \Rightarrow 3 P^2=-6 P^2 \cos \theta \\
& \Rightarrow \cos \theta=-\frac{1}{2} \text { or } \theta=120^{\circ}
\end{aligned}$

Hence, the answer is the option (1).

Example 2: The ratio of maxium and minimum magnitudes of the resultant of two vectors \text { }|\vec{a}| \text { and }|\vec{b}| is 3:1, Now |\vec{a}|=

1) $\mid \vec{b}$
2) ($2 \mid \vec{b}$
3) $3 \mid \vec{b}$
4) $4|\vec{b}|$

Solution:

$
\begin{aligned}
& |\vec{a}+\vec{b}|=\sqrt{a^2+b^2+2 a b \cos \theta} \\
& |\vec{a}+\vec{b}|_{\max }=a+b{ }_{\text {when } \theta=0^{\circ}} \\
& |\vec{a}+\vec{b}|_{\min }=a-b \quad \text { when } \theta=180^{\circ}
\end{aligned}
$
here
$
\frac{a+b}{a-b}=\frac{3}{1}
$
or
$
a+b=3 a-3 b
$
or $2 a=4 b \Rightarrow a=2 b$
$
|\vec{a}|=2|\vec{b}|
$

Hence, the answer is option (2).

Example 3:

Two vectors $\vec{A}$ and $\vec{B}$ have equal magnitudes. The magnitudes of $(\vec{A}+\vec{B})$ is 'n' times the magnitudes of $(\vec{A}-\vec{B})$. The angle between $\vec{A}$ and $\vec{B}$ is :

1) $
\sin ^{-1}\left[\frac{n-1}{n+1}\right]
$
2) $ \cos ^{-1}\left[\frac{n^2-1}{n^2+1}\right]
$
3) $ \sin ^{-1}\left[\frac{n^2-1}{n^2+1}\right]
$
4) $\cos ^{-1}\left[\frac{n-1}{n+1}\right]
$

Solution:

$\begin{aligned}
& A^2+B^2+2 A B \cos \theta=n^2\left(A^2+B^2-2 A B \cos \theta\right) \\
& \because|\vec{A}|=|\vec{B}| \\
& \therefore A^2+A^2+2 A^2 \cos \theta=n^2\left(A^2+A^2-2 A^2 \cos \theta\right) \\
& 2 A^2(1+\cos \theta)=2 A^2 n^2(1-\cos \theta) \\
& \cos \theta\left(1+n^2\right)=n^2-1 \\
& \cos \theta=\frac{n^2-1}{n^2+1} \\
& \theta=\cos ^{-1}\left(\frac{n^2-1}{n^2+1}\right)
\end{aligned}$

Hence, the answer is the Option (2)

Example 4:

The magnitude and direction of the resultant of two vectors $\vec{A}$ and $\vec{B}$ in terms of their magnitudes and angle $\theta$ between them is :

1) $
R=\sqrt{A^2+B^2-2 A B \cos \theta}, \tan \alpha=\frac{B \sin \theta}{A+B \cos \theta}
$
2) $
R=\sqrt{A^2+B^2+2 A B \cos \theta}, \tan \alpha=\frac{B \cos \theta}{A+B \sin \theta}
$
3) $
R=\sqrt{A^2+B^2-2 A B \cos \theta}, \tan \alpha=\frac{A \sin \theta}{B+A \sin \theta}
$
4) $
R=\sqrt{A^2+B^2+2 A B \cos \theta}, \tan \alpha=\frac{B \sin \theta}{A+B \cos \theta}
$

Solution:

In the Parallelogram law of vector addition

The figure given below represents the law of parallelogram vector Addition.

$\vec{R}=\vec{A}+\vec{B}$

Using Pythagorean theorem in triangle MOP-

$\begin{aligned}
& R^2=(B \sin \Theta)^2+(A+B \cos \Theta)^2 \\
\Rightarrow & R=\sqrt{A^2+B^2+2 A B \cos \Theta} \\
\Rightarrow & \tan \alpha=\frac{B \sin \Theta}{A+B \cos \Theta}
\end{aligned}$
Hence, the answer is option (4).

Example 5: Two forces P and Q, of magnitude 2F and 3F, respectively, are at an angle $\theta$ with each other. If the force Q is doubled, then their resultant also gets doubled. Then the angle $\theta$ (in degrees) is :

1) 0

2) 120

3) 90

4) 110

Solution:

Parallelogram law of vector addition -

- wherein

$
R^2=P^2+Q^2+2 P Q \cos \theta
$

Represents the law of parallelogram Vector Addition
Case 1
$
R^2=4 F^2+9 F^2+12 F^2 \cos \theta
$

Case 2
$
\begin{aligned}
& 4 R^2=4 F^2+36 F^2+24 F^2 \cos \theta \\
& 16 F^2+36 F^2+48 F^2 \cos \theta=4 F^2+36 F^2+24 F^2 \cos \theta \\
& \therefore 12 F^2=-24 F^2 \cos \theta \\
& \theta=120^{\circ}
\end{aligned}
$
Hence, the answer is option (2).

Summary

Understanding vector addition and subtraction is crucial for accurately analyzing physical phenomena involving directional quantities. By mastering the methods for combining vectors, whether in the same or different directions, one can effectively apply these concepts to solve complex problems in physics and engineering, utilizing the triangle and parallelogram laws of vector addition

Frequently Asked Questions (FAQs)

1. What is a vector in physics?

A vector in physics is a quantity that has both magnitude and direction, represented by an arrow.

2. What is a unit vector?

A unit vector is a vector with a magnitude of one. It indicates direction only.

3. How do you multiply vectors?

Vectors can be multiplied in several ways: scalar multiplication (multiplying a vector by a scalar quantity), dot product (or scalar product, yielding a scalar), and cross product (or vector product, yielding another vector).

4. When do you use vector components?

Vector components are used to simplify vector calculations, especially in coordinate systems like Cartesian coordinates. They break down a vector into its perpendicular components along the coordinate axes (x, y, z).

5. How does vector addition apply to forces in physics?

In physics, forces are vector quantities. When multiple forces act on an object, the net force is determined by vector addition of all individual forces. This principle is crucial in understanding the resultant motion of objects under the influence of multiple forces, as described by Newton's laws of motion.

6. What is the concept of equilibrium in terms of vector addition?

In physics, equilibrium occurs when the vector sum of all forces acting on an object is zero. This means that the forces balance each other out, resulting in no net force and no acceleration. Understanding vector addition is crucial for analyzing equilibrium situations in mechanics and statics.

7. How does vector addition apply to momentum conservation?

The law of conservation of momentum states that the total momentum of a closed system remains constant. In collisions or explosions, the vector sum of momenta before the event equals the vector sum after the event. This principle relies on vector addition to account for both magnitude and direction of momenta.

8. What is the triangle inequality theorem in vector addition?

The triangle inequality theorem states that the magnitude of the sum of two vectors is always less than or equal to the sum of the magnitudes of the individual vectors. Mathematically, |A + B| ≤ |A| + |B|. The equality holds only when the vectors are parallel and in the same direction.

9. What is the concept of vector resolution, and how does it relate to addition?

Vector resolution is the process of breaking down a vector into its components along chosen axes. It's the inverse of vector addition. By resolving vectors into components, complex addition problems can be simplified to adding scalar quantities along each axis. This is particularly useful when dealing with non-perpendicular vectors or multiple vectors.

10. What is the commutative property of vector addition?

The commutative property of vector addition states that the order in which vectors are added does not affect the final result. Mathematically, it can be expressed as A + B = B + A, where A and B are vectors. This property holds true for both two-dimensional and three-dimensional vectors.

11. Can you explain the parallelogram method of vector addition?

The parallelogram method is a graphical technique for adding two vectors. To use this method, draw the two vectors with a common starting point, then complete the parallelogram using these vectors as sides. The diagonal of the parallelogram from the common starting point represents the resultant vector's magnitude and direction.

12. How do you add perpendicular vectors?

To add perpendicular vectors, you can use the Pythagorean theorem. If vectors A and B are perpendicular, the magnitude of the resultant vector R is given by |R| = √(|A|² + |B|²). The direction of R can be found using the inverse tangent function: θ = tan⁻¹(|B|/|A|), where θ is the angle between R and A.

13. How does the magnitude of the resultant vector relate to the magnitudes of the added vectors?

The magnitude of the resultant vector is not always equal to the sum of the magnitudes of the added vectors. It depends on the angle between the vectors. The resultant magnitude is only equal to the sum of the individual magnitudes when the vectors are parallel and in the same direction. In all other cases, the resultant magnitude is less than the sum of the individual magnitudes.

14. What is the difference between collinear and parallel vectors in addition?

Collinear vectors lie on the same line and can be either parallel (pointing in the same direction) or antiparallel (pointing in opposite directions). Parallel vectors have the same direction but may not lie on the same line. When adding collinear vectors, the result is always a vector on the same line. For parallel vectors, the result may not lie on the same line as the original vectors.

15. How do you add vectors in three dimensions?

To add vectors in three dimensions, you add the corresponding components of each vector. If A = (Ax, Ay, Az) and B = (Bx, By, Bz), then A + B = (Ax + Bx, Ay + By, Az + Bz). The magnitude of the resultant vector can be found using the three-dimensional Pythagorean theorem: |R| = √(Rx² + Ry² + Rz²).

16. What is the concept of vector resolution in polar coordinates?

Vector resolution in polar coordinates involves expressing a vector in terms of its magnitude and direction angle. While addition in polar form is more complex than in Cartesian coordinates, understanding this concept is crucial for problems involving circular motion, planetary orbits, and other scenarios where polar coordinates are more natural.

17. What is the tip-to-tail method of vector addition?

The tip-to-tail method, also known as the triangle method, is a graphical technique for adding vectors. In this method, you draw the first vector, then draw the second vector starting from the tip of the first vector. The resultant vector is drawn from the tail of the first vector to the tip of the last vector.

18. How does vector addition apply to electrical circuits?

In electrical circuits, vector addition is used when dealing with alternating current (AC) circuits. Voltages and currents in AC circuits are represented as vectors (or phasors), and their addition takes into account both magnitude and phase angle. This is crucial for analyzing complex circuits and power systems.

19. How does vector addition apply to wave superposition?

In wave physics, the principle of superposition states that when two or more waves overlap, the resultant displacement at any point is the vector sum of the displacements of the individual waves. This applies to all types of waves, including sound, light, and water waves, and is crucial in understanding interference patterns.

20. What is the relationship between vector addition and the parallelogram law of forces?

The parallelogram law of forces is a direct application of vector addition. It states that when two forces act on a point, their resultant is represented in magnitude and direction by the diagonal of the parallelogram formed by the two forces. This law is fundamental in mechanics and is essentially a graphical representation of vector addition.

21. What is the concept of vector projection, and how does it relate to addition?

Vector projection involves finding the component of one vector along the direction of another. While not directly a part of vector addition, projections are often used to simplify addition problems. By projecting vectors onto common axes, complex addition scenarios can be reduced to simpler component-wise additions.

22. What is vector addition?

Vector addition is the process of combining two or more vectors to form a single resultant vector. It takes into account both the magnitude and direction of the vectors being added. The result of vector addition is represented graphically by the tip-to-tail method or parallelogram method.

23. How does vector addition differ from scalar addition?

Vector addition differs from scalar addition in that it considers both magnitude and direction, while scalar addition only deals with magnitudes. When adding vectors, we must account for their directions, which can result in partial cancellation or reinforcement of the vectors.

24. How do you add three or more vectors?

To add three or more vectors, you can use either the tip-to-tail method or algebraic components. In the tip-to-tail method, draw the vectors sequentially, with each new vector starting at the tip of the previous one. The resultant is the vector from the tail of the first to the tip of the last. Algebraically, add the x-components and y-components separately, then combine them to find the resultant.

25. How does vector addition apply to displacement in physics?

In physics, displacement is a vector quantity representing the shortest distance and direction from the initial to the final position. When an object undergoes multiple displacements, the total displacement is found by vector addition of the individual displacements. This allows us to determine the net change in position, considering both distance and direction.

26. What is the role of unit vectors in vector addition and subtraction?

Unit vectors are vectors with a magnitude of 1 and are often used to represent direction. In vector addition and subtraction, unit vectors (usually denoted as i, j, and k for the x, y, and z directions) help express vectors in component form. This makes it easier to perform calculations, especially in three dimensions.

27. What is vector subtraction?

Vector subtraction is the process of finding the difference between two vectors. It can be thought of as adding the negative of a vector. Mathematically, A - B is equivalent to A + (-B), where -B is the vector with the same magnitude as B but pointing in the opposite direction.

28. How does changing the order of vectors affect subtraction?

Unlike addition, vector subtraction is not commutative. Changing the order of vectors in subtraction will result in a different answer. For example, A - B ≠ B - A. The result will have the same magnitude but point in opposite directions.

29. What is the graphical representation of vector subtraction?

Graphically, vector subtraction can be represented by drawing the two vectors with a common starting point, then drawing a vector from the tip of the subtrahend (the vector being subtracted) to the tip of the minuend (the vector being subtracted from). This resulting vector represents the difference.

30. How do you subtract vectors using components?

To subtract vectors using components, you subtract the corresponding components of the vectors. If A = (Ax, Ay) and B = (Bx, By), then A - B = (Ax - Bx, Ay - By). After subtracting the components, you can find the magnitude and direction of the resultant vector using the Pythagorean theorem and inverse tangent function.

31. What is the zero vector in vector addition and subtraction?

The zero vector is a vector with magnitude zero and no defined direction. In vector addition, adding the zero vector to any other vector leaves that vector unchanged. In subtraction, subtracting a vector from itself results in the zero vector.

32. What is the significance of the dot product in vector addition?

While not directly used in vector addition, the dot product is an important operation in vector algebra. It gives a scalar result and is used to find the angle between vectors or project one vector onto another. Understanding the dot product can help in decomposing vectors for addition and in analyzing the work done by forces.

33. How does vector subtraction relate to relative motion?

Vector subtraction is essential in understanding relative motion. The relative velocity of object A with respect to object B is found by subtracting the velocity vector of B from the velocity vector of A. This concept is crucial in many areas of physics, including kinematics and relativity.

34. What is the concept of vector decomposition, and how does it relate to addition?

Vector decomposition is the process of breaking down a vector into components along specified directions. It's the inverse of vector addition and is crucial for solving complex problems. By decomposing vectors, you can add or subtract their components separately, making calculations more manageable, especially when dealing with non-perpendicular vectors.

35. How do you add vectors of different units?

Vectors can only be added if they represent the same physical quantity and have the same units. You cannot directly add, for example, a velocity vector to a force vector. In cases where different vector quantities interact (like force and displacement in work calculations), other vector operations like the dot product are used instead of direct addition.

36. How does vector subtraction apply to finding relative position?

Relative position can be determined using vector subtraction. If you have position vectors for two objects, A and B, the position of B relative to A is found by subtracting A's position vector from B's. This concept is crucial in astronomy, navigation, and any field dealing with relative spatial relationships.

37. What is the significance of the cross product in relation to vector addition?

While the cross product is not directly used in vector addition, it's an important operation in vector algebra. The cross product of two vectors results in a vector perpendicular to both, which is useful in calculating torque, angular momentum, and areas. Understanding the cross product complements the concept of vector addition in solving complex physics problems.

38. What is the concept of a null vector, and how does it affect addition?

A null vector, also known as a zero vector, is a vector with zero magnitude and undefined direction. In vector addition, adding a null vector to any other vector leaves that vector unchanged. This property is similar to adding zero in scalar arithmetic and is important in understanding vector spaces and linear algebra.

39. How does vector addition apply to center of mass calculations?

The center of mass of a system can be calculated using vector addition. Each particle's position is multiplied by its mass (creating a moment vector), and these vectors are added. The sum is then divided by the total mass of the system. This concept is crucial in mechanics, especially in analyzing the motion of complex systems.

40. What is the difference between scalar multiplication and vector addition?

Scalar multiplication involves multiplying a vector by a scalar (a regular number), which changes the vector's magnitude and potentially its direction (if the scalar is negative). Vector addition, on the other hand, combines two or more vectors to produce a new vector. Both operations are fundamental in vector algebra but serve different purposes.

41. How does vector addition apply to calculating gravitational fields?

The gravitational field at a point due to multiple masses is found by vector addition of the individual gravitational field vectors produced by each mass. This principle is crucial in astrophysics for understanding the gravitational effects of planetary systems, galaxies, and other celestial bodies.

42. What is the concept of a position vector, and how does it relate to vector addition?

A position vector is a vector that represents the position of a point relative to a chosen origin. The displacement between two points can be found by subtracting their position vectors. This application of vector addition and subtraction is fundamental in kinematics and coordinate geometry.

43. How does vector addition apply to electromagnetic fields?

In electromagnetism, the total electric or magnetic field at a point due to multiple sources is found by vector addition of the individual fields produced by each source. This principle, known as superposition, is crucial for analyzing complex electromagnetic systems and understanding phenomena like interference and diffraction.

44. What is the concept of a unit vector, and how is it used in vector addition?

A unit vector is a vector with a magnitude of 1 and is often used to specify direction. In vector addition, unit vectors (usually denoted as î, ĵ, and k̂ for the x, y, and z directions) are used to express vectors in component form. This makes it easier to perform calculations, especially in three dimensions.

45. How does vector addition apply to momentum in quantum mechanics?

In quantum mechanics, momentum is treated as a vector quantity. The total momentum of a system of particles is the vector sum of the individual momenta. This concept is important in understanding conservation laws at the quantum level and in analyzing particle interactions.

46. What is the concept of vector components, and how do they simplify addition?

Vector components are the projections of a vector onto the coordinate axes. Breaking vectors into components simplifies addition by allowing you to add corresponding components separately. This method is particularly useful when dealing with non-perpendicular vectors or multiple vectors in complex arrangements.

47. How does vector addition apply to calculating torque?

While torque itself is calculated using the cross product, vector addition is used when multiple torques act on a system. The net torque is the vector sum of all individual torques. This principle is crucial in rotational dynamics and the analysis of equilibrium in static systems.

48. What is the concept of a resultant vector, and how is it determined?

A resultant vector is the single vector that represents the combined effect of two or more vectors. It is determined through vector addition, either graphically (using methods like the parallelogram or tip-to-tail) or algebraically (by adding vector components). The resultant summarizes the overall effect of multiple vectors acting together.

49. How does vector addition apply to calculating electric potential?

While electric potential is a scalar quantity, it's calculated from the electric field, which is a vector. The total electric potential at a point due to multiple charges is the scalar sum of the potentials due to each charge. This involves vector operations in the calculation of the electric field before summing the potentials.

50. How does vector addition apply to analyzing complex motion?

In analyzing complex motion, like projectile motion or circular motion, vector addition is used to combine different components of motion. For example, in projectile motion, the horizontal and vertical components of velocity are treated separately and then combined using vector addition to determine the overall motion of the projectile.

51. What is the significance of the associative property in vector addition?

The associative property of vector addition states that (A + B) + C = A + (B + C), where A, B, and C are vectors. This property allows flexibility in the order of adding multiple vectors, which is particularly useful in simplifying complex addition problems or in proving vector relationships.

52. How does vector addition apply to analyzing magnetic flux?

Magnetic flux is a scalar quantity, but it's calculated using the dot product of the magnetic field vector and the area vector. When dealing with multiple magnetic fields or complex surfaces, vector addition of the magnetic field vectors is necessary before calculating the flux. This concept is crucial in electromagnetism and electrical engineering.

53. What is the concept of a displacement vector, and how does it relate to vector addition?

A displacement vector represents the change in position from an initial point to