What is Vector Addition?
The addition of Physical quantities through Mathematical operations is called Vector Addition. Vector Addition involves only the vector quantities and not the scalar quantities. Vector quantities are added to determine the resultant direction and magnitude of a quantity.
According to Newton's law of motion, the net force acting on an object is calculated by the vector sum of individual forces acting on it.
The net force is the resultant of the addition of all force vectors. The rules of Vector Addition are elementary. Try to observe the addition of the following force vectors.
\[\overrightarrow{10}\] + \[\overrightarrow{10}\] =\[\overrightarrow{20}\]
\[\overrightarrow{10}\] + \[\overrightarrow{5}\] =\[\overrightarrow{15}\]
\[\overrightarrow{10}\] + \[\overrightarrow{10}\] =\[\overrightarrow{0}\]
\[\overrightarrow{10}\] + \[\overrightarrow{-15}\] =\[\overrightarrow{-5}\]
\[\uparrow{20}\] + \[\downarrow{15}\] =\[\uparrow{5}\]
\[\overrightarrow{10}\] + \[\overrightarrow{-15}\] =\[\overrightarrow{-5}\]
Parallelogram Law of Vectors
State parallelogram law of Vector Addition- As per this law, the summation of squares of lengths of four sides of a parallelogram equals the summation of squares of length of the two diagonals of the parallelogram.
In Euclidean geometry, a parallelogram must be opposite sides and of equal length.
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ABCD is a parallelogram, where AB = DC and AD = BC.
As per the law,
2(AB)2 + 2 (BC)2 = (AC)2 + (BD)2
If the parallelogram is a rectangle, then it can be written as,
2(AB)2 + 2 (BC)2 = 2(AC)2
Because, in rectangle, the two diagonals are of equal lengths
i.e., (AC = BD)
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If two vectors act on a single point simultaneously, then the magnitude and direction of the resultant vector are drawn by the adjacent sides of the point. Therefore, the resultant vector is represented both in direction and magnitude by the diagonal vector of the parallelogram, which passes through the point.
Consider the above figure,
In the above figure, the vector P and the vector Q represent the sides, OA and OB, respectively.
As per the law, the side OC of the parallelogram will represent the resultant vector R.
OA OB=OC (or)
\[\overrightarrow{P}\] + \[\overrightarrow{Q}\] =\[\overrightarrow{R}\]
Parallelogram Law of Vector Addition
If two vectors that are simultaneously acting on a point, represented by the adjacent sides of the parallelogram, which are drawn from the point, then the resultant vector is represented by the diagonal of the parallelogram that passes through that point. The resultant vector represents both magnitude and direction.
Proof:
Let AD = BC = x, and AB = DC = y, and ∠ BAD = α
Using the law of cosines in triangle BAD, we get
x2 + y2 – 2xy cos (α) = BD2-------------(1)
Since adjacent angles are supplementary,
So,
∠ADC = 180 – α
Here, using the law of cosines in triangle ADC, we get
x2 + y2 – 2xy cos (180 – α) = AC2 --------------(2)
We know that cos(180 – x) = – cos x in (2)
Applying it in eqn----(2)
x2 + y2 + 2xy cos(α) = AC2
Now, adding the eq (1) and eq (2) (BD2 + AC2), we get
BD2 + AC2 = x2 + y2 – 2xy cos(α) + x2 + y2 + 2xycos(α)
After simplifying the above expression, we get
BD2 + AC2 =2x2 + 2 y2--------(3)
The above equation can also be written as:
BD2 + AC2 = 2(AB)2 + 2( BC)2
This proves the parallelogram law.
Vector Addition Analytical Method
Analytical methods of Vector Addition and vector subtraction use geometry and trigonometry. It also uses some parts of graphical techniques because vectors are represented as arrows for visualization.
Analytical methods are more concise, precise, and more accurate as compared to graphical methods. The accuracy of the graphical method is limited due to drawings that can be drawn. The only limitation of analytical methods is the precision and accuracy of Physical quantities.
Resolving a Vector into Perpendicular Components
Analytical techniques and right triangles are useful to calculate Physical parameters because motions of particles in the perpendicular directions are independent. Almost every time, a vector is separated into perpendicular components.
For example, given a vector-like AA in the below Figure, there are two perpendicular vectors, Ax and Ay, which add up to produce a resultant vector A.
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The vector A originates from the origin of a xy-coordinate system with its x and y components as Ax and Ay, respectively, as shown in the figure above. These vectors form a right-angled triangle. The analytical relationship among these vectors is mentioned below.
Ax = component of A vector along x-axis.
Ay = component of A vector along y-axis.
The three vectors A, Ax, and Ay form a right-angled triangle.
Also
Ax + Ay = A.
This relationship between components of the vector and resultant vectors is only for vector quantities and not for scalar quantities.
For example, if Ax = 6m towards east, Ay = 8 m towards north, and A = 10 m towards north-east, then the relation of vector Ax + Ay = A. However, the sum of magnitudes of the vectors will not be equal. That is,
6 m + 8 m ≠ 10 m
Also,
Ax + Ay ≠ A
If the vector AA is known, then its magnitude A and direction θ is also known. To find Ax and Ay by its x and y components, the following relationships of the right-angled triangle are used.
Ax = Acosθ, and
Ay = Asinθ
Importance of Vector in Physics
Many values used in Physics are vectors, therefore understanding vectors is essential. If you try to combine vector values without considering their direction, you will receive the wrong result. Vectors merely decrease the number of equations that must be solved. We could have done our Physics without vectors, but it would have been more difficult. They are Mathematical tools that help us manage equations and calculations more efficiently and effectively. Physical quantities can be represented using vectors. Vectors are most commonly used in Physics to describe displacement, velocity, and acceleration. Vectors are arrows that represent a combination of magnitude and direction. The magnitude is represented by the length, and the direction of that value is the direction in which the vector points. Vectors are useful in Physics because they may graphically express location, velocity, displacement, and acceleration. Vector Addition is significant in engineering since it involves forces, electric fields, magnetic fields, velocities, position, polarization, momentum, angular momentum, trajectories, current density, magnetization, and so on.
Adding Vectors Graphically
Vectors can be graphically added by arranging them end to end on a set of axes. The simplicity with which vectors may be added to one another is one of the ways in which presenting Physical values as vectors simplifies analysis. Because vectors are graphical representations, Vector Addition may be done graphically. The head-to-tail method is another name for the graphical method of Vector Addition. To begin, create a set of coordinate axes. After that, create the first vector with its base at the origin of the coordinate axes. Because Vector Addition is commutative, it makes no difference which vector is drawn first. The next step is to construct the next vector such that its tail begins at the previous vector's head (the arrow side). Continue to insert each vector at the head of the one before it until all of the vectors you want to add are connected. Finally, draw a straight line from the origin to the head of the chain's final vector. This new line represents the vector outcome of combining those vectors.
Characteristics of Vectors
Vectors have the following characteristics:
The Vectors have both magnitude as well as direction.
These change when either the magnitude or the direction, or both, change.
A vector is defined by its magnitude and direction but not by its position. To put it another way, if a vector is moved parallel to itself, its length remains the same.
Vectors are distinguished by the use of bold-faced letters or letters with an arrow over them.
The negative vector is a vector that has the inverse direction of the reference vector.
A vector is often represented as an arrow with the same direction as the quantity and a length proportionate to the magnitude of the quantity.
Many other Physical variables are described by vectors, including displacement, linear displacement, linear momentum, linear acceleration, angular acceleration, and angular momentum.
Vector Application in Real Life
In Physics, vectors are used to describe anything that has both a direction and a magnitude. They are often represented by pointing arrows, the length of which denotes the magnitude of the vector.
Vectors may represent a wide range of physical objects and events. Wind, for example, is a vector quantity since it has a direction and a magnitude at any given position. By drawing wind vectors for a variety of geographic areas, you might build a map of airflow at any moment in time.
Many of the characteristics of moving objects are vectors as well. Consider the motion of a billiard ball over a table. The velocity vector of a ball defines its motion—the direction of the vector arrow indicates the ball's motion, and the length of the vector indicates the ball's speed.
Consider the forces at work on a river-crossing boat. The boat's motor generates force in one direction, while the river current generates force in the other. Both forces are represented by vectors.
FAQs on Vector Addition
1. What are the main methods for adding two or more vectors?
There are two primary methods for adding vectors as per the CBSE Class 11 syllabus for 2025-26:
- Graphical Methods: These methods use geometric constructions to find the resultant vector. The main graphical methods are the Triangle Law of Vector Addition and the Parallelogram Law of Vector Addition. They are useful for visualising the result.
- Analytical Method: This method uses trigonometry and vector components to calculate the precise magnitude and direction of the resultant vector. It is more accurate than graphical methods and is essential for solving complex physics problems.
2. What is the Parallelogram Law of Vector Addition?
The Parallelogram Law of Vector Addition states that if two vectors acting simultaneously at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from that point, then their resultant vector is represented both in magnitude and direction by the diagonal of the parallelogram passing through that same point.
3. What is the Triangle Law of Vector Addition?
The Triangle Law of Vector Addition provides another way to find the resultant of two vectors. It states that if two vectors can be represented both in magnitude and direction by two sides of a triangle taken in the same order, then their resultant is represented completely, both in magnitude and direction, by the third side of the triangle taken in the opposite order.
4. How is the resultant of two vectors calculated using the analytical method?
Using the analytical method, the magnitude (R) of the resultant vector of two vectors A and B with an angle θ between them is given by the formula: R = √(A² + B² + 2ABcosθ). The direction of the resultant vector can be found using the angle (α) it makes with vector A, calculated by: tan α = (Bsinθ) / (A + Bcosθ). This method gives a precise mathematical solution.
5. Why is the concept of vector addition so important in Physics?
Vector addition is crucial because many fundamental physical quantities are vectors, such as force, velocity, acceleration, and displacement. To understand the net effect of multiple forces acting on an object or the resultant velocity of a boat crossing a river, we must use vector addition. It allows us to accurately predict the outcome of interactions involving quantities that have both magnitude and direction.
6. How does vector addition differ from simple scalar addition?
The key difference lies in the treatment of direction. Here's a comparison:
- Scalar Addition: Involves adding only magnitudes. It follows simple arithmetic rules. For example, a mass of 5 kg plus a mass of 2 kg is always 7 kg.
- Vector Addition: Involves adding quantities that have both magnitude and direction. The resultant depends on the angle between the vectors. For example, a force of 5 N plus a force of 2 N can result in any value between 3 N (if they are in opposite directions) and 7 N (if they are in the same direction).
7. Can the sum of two vectors of unequal magnitude ever be zero?
No, the sum of two vectors with unequal magnitudes can never be zero. A resultant of zero (a null vector) can only be achieved if two vectors have the exact same magnitude and act in exactly opposite directions (i.e., the angle between them is 180°). The sum of three or more vectors of unequal magnitudes can, however, be zero if they form a closed polygon when arranged head-to-tail.
8. What are the fundamental properties that govern vector addition?
Vector addition follows two fundamental properties that are essential for calculations in physics:
- Commutative Law: This property states that the order in which vectors are added does not affect the result. Mathematically, A + B = B + A.
- Associative Law: This property states that when adding three or more vectors, the way they are grouped does not change the final resultant. Mathematically, (A + B) + C = A + (B + C).
9. How does resolving a vector into components simplify the process of addition?
Resolving vectors into their perpendicular components (e.g., along the x and y-axes) is a powerful technique that simplifies addition, especially for multiple vectors. Instead of complex geometric constructions, you can:
- Break down each vector into its x and y components.
- Add all the x-components together algebraically to get the total x-component of the resultant (Rx).
- Add all the y-components together to get the total y-component (Ry).
- Combine Rx and Ry using the Pythagorean theorem to find the final resultant's magnitude.
This transforms a complex geometry problem into a straightforward algebraic calculation.
