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, A_x and A_y, 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 A_x and A_y, respectively, as shown in the figure above. These vectors form a right-angled triangle. The analytical relationship among these vectors is mentioned below.

A_x = component of A vector along x-axis.

A_y = component of A vector along y-axis.

The three vectors A, A_x, and A_y form a right-angled triangle.

Also

A_x + A_y = A.

This relationship between components of the vector and resultant vectors is only for vector quantities and not for scalar quantities.

For example, if A_x = 6m towards east, A_y = 8 m towards north, and  A = 10 m towards north-east, then the relation of vector A_x + A_y = A. However, the sum of magnitudes of the vectors will not be equal. That is,

6 m + 8 m ≠ 10 m

Also,

A_x + A_y ≠ A

If the vector AA is known, then its magnitude A and direction θ is also known. To find A_x and A_y by its x and y components, the following relationships of the right-angled triangle are used.

A_x = Acosθ, and

A_y = 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.