Vector Addition

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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.

10+ 10=20

10+ 5 = 15

10+ 10=0

10+ -15 = -5

20+ 15=5

10+ -5=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 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)

P+Q= 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 pass 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 – 2xycos (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 – 2xycos (180 – α) = AC--------------(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

BD+ AC= x2 + y2 – 2xycos(α) + x2 + y2 + 2xycos(α)

After simplifying the above expression, we get

BD+ AC=2x2 + 2 y2--------(3)

The above eqn can also be written as:

BD+ AC= 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 is originated 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

FAQ (Frequently Asked Questions)

Q1. Write Some Applications of Parallelogram Law of Vectors.

Ans- It is used to find the resultant of two vector quantities like force and velocity. 

However, the parallelogram law of vector addition is not used to find resultant scalar quantities like energy, work, and speed, rather simple arithmetic is used to do so.

Q2. Why Vector Addition is Important.

Ans- Knowledge of vectors is important because there is a large number of physical quantities that have both magnitude and direction. 

Vector quantities are added by keeping their magnitude and direction in account. Some of the major vector quantities in physics are force, velocity, acceleration, and displacement.

Q3. What are the Characteristics of Vector Addition?

Ans- The addition of vectors should satisfy two important properties.

  • The commutative law: It states the order in which the vectors are added doesn't matter: a + b = b + a.

  • The associative law: It states that the sum of multiple vectors does not depend on the pair of vectors which is added first: (a + b) + (c + d) = (a + d) + (b + c).

Q4. Practice Questions on Vector Addition. Add the Given Vectors & Identify the Resultant.

Ans- Vector 1: magnitude = 3.0 m/s and direction = 450

Vector 2: magnitude = 5.0 m/s and direction = 1350

vector 1 + vector 2 = 5.83 m/s, and direction = 1040 

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