20 Physics -- Vectors

Proof Parallelogram Law

1 Answers

Parallelogram Law of Vector Addition Formula

Consider two vectors P and Q with an angle θ between them. The sum of vectors P and Q is given by the vector R, the resultant sum vector using the parallelogram law of vector addition. If the resultant vector R makes an angle ϕ with the vector P, then the formulas for its magnitude and direction are:

|R| = √(P² + Q² + 2PQ cos θ)
β = tan^-1[(Q sin θ)/(P + Q cos θ)]

Parallelogram Law of Vector Addition Proof

Let us first see the statement of the parallelogram law of vector addition:

Statement of Parallelogram Law of Vector Addition: If two vectors can be represented by the two adjacent sides (both in magnitude and direction) of a parallelogram drawn from a point, then their resultant sum vector is represented completely by the diagonal of the parallelogram drawn from the same point.

Now, to prove the formula of the parallelogram law, we consider two vectors P and Q represented by the two adjacent sides OB and OA of the parallelogram OBCA, respectively. The angle between the two vectors is θ. The sum of these two vectors is represented by the diagonal drawn from the same vertex O of the parallelogram, the resultant sum vector R which makes an angle β with the vector P.

parallelogram law of vector addition proof

Extend the vector P till D such that CD is perpendicular to OD. Since OB is parallel to AC, therefore the angle AOB is equal to the angle CAD as they are corresponding angles, i.e., angle CAD = θ. Now, first, we will derive the formula for the magnitude of the resultant vector R (side OC).

In right-angled triangle OCD, we have

OC² = OD² + DC²

⇒ OC² = (OA + AD)² + DC² --- (1)

In the right triangle CAD, we have

cos θ = AD/AC and sin θ = DC/AC

⇒ AD = AC cos θ and DC = AC sin θ

⇒ AD = Q cos θ and DC = Q sin θ --- (2)

Substituting values from (2) in (1), we have

R² = (P + Q cos θ)² + (Q sin θ)²

⇒ R² = P² + Q²cos²θ + 2PQ cos θ + Q²sin²θ

⇒ R² = P² + 2PQ cos θ + Q²(cos²θ + sin²θ)

⇒ R² = P² + 2PQ cos θ + Q² [cos²θ + sin²θ = 1]

⇒ R = √(P² + 2PQ cos θ + Q²) → Magnitude of the resultant vector R

Next, we will determine the direction of the resultant vector. We have in right traingle ODC,

tan β = DC/OD

⇒ tan β = Q sin θ/(OA + AD) [From (2)]

⇒ tan β = Q sin θ/(P + Q cos θ) [From (2)]

⇒ β = tan^-1[(Q sin θ)/(P + Q cos θ)] → Direction of the resultant vector R

The resultant of two equal forces may have a magnitude equal toone of the forces. At what angle between the two equal forces,this is possible? Justify your answer.

Let the forces be F1 and F2 at an angle A and their resultant be F. According to question,

| F1 | = | F2 | = | F |

We know,

F^2 = F1^2 + F2^2 + 2 F1 F2 cos A

or, F^2 = F^2 + F^2 + 2 F F cos A

or, F^2 - 2F^2 = 2 F^2 cos A

or, - F^2 = 2 F^2 cos A

or, -1 = 2 cos A

or, cos A = -1/2

or, cos A = cos 120

So, A = 120 degrees.

Hence, when two forces of equal magnitude act at an angle of 120 degrees, the resultant is equal to one of the forces.

Is Physical quantity having magnitude and direction necessarily avector quantity? Explain.

No, it is not necessary that a physical quantity that has magnitude and direction be a vector quantity.But yes a vector quantity should have magnitude and direction and apart from that, it should also follow vector laws of addition. One eg. for a quantity that has magnitude and direction but does not follow vector laws of addition is electric current.

Can a physical quantity have direction and still be a scalar?

Yes, a physical quantity can have magnitude and direction but still be a scalar if it doesn't obey the vector addition. An example is Electric Current which has magnitude and a fixed direction, but it does not follow vector laws of addition.

How many components can be realized of a vector?

Any vector directed in two dimensions can be thought of as havingtwo different components. The component of a single vector describes the influence of that vector in a given direction.

Is it possible that the resultant of two vectors be smaller than the smaller of the two vectors? If so, under what condition?

Yes, if the angle between the two vectors is more than 90^obut less than 270⁰. (cosΘ is negative)

What happens when a vector is multiplied by a real number?

If a vector A is multiplied by a real number (say n), the vector of same nature is obtained but its magnitude is n times that of A.

Can two vectors of different magnitude be combined to give a zero resultant? What about in three vectors?

The two vectors (say A and B) of different magnitudes cannot be combined togive zero resultant since minimum value of combination isІA-BІ which is not zero if AB.

The three vectors A, B and Cof different magnitudes can be zero such that they form a closed triangle, then,

A+B+C=0

or, C=-(A+B)

Hence, the sum of three vectors may be zero if vector sum of any two vectors is equal and opposite to the third vector.

Note: The vectors can give this result only if they lie in the same plane. If three...

Does it have any sense that the vector has zero magnitudes?

Yes, a vector which has zero magnitude is also a vector in case of two vectors travelling in opposite directions with equal magnitudes. At this case, the resultant vector has zero magnitude but it is still a vector. We call it a null vector.

Can three vectors not lying in a plane give zero resultant?

No, if three vectors do not lie in a plane, they cannot give zero resultant.

Explanation:

Let A, B and Cbe three vectors. If they give zero resultant, then

A+B+C=0

or, A= -(B+C)

Hence, they will produce zero resultant, if Ais equal to negative of vector (B+C). The vector (B+C) lies in the plane of B and C.Hence, Awill be equal to negative of (B+C) if A,B and C all lie in a plane.

What does the negative of a vector mean?

A negative of a vector represents the direction opposite to the reference direction. It means that the magnitude of two vectors are same but they are opposite in direction.

For example, if A and B are two vectors that have equal magnitude but opposite in direction, then vector A is negative of vector B.

Torque and force are two different physical quantities, can they be added?

Torque has the dimension of force times distance, symbolically T²L²M. Although those fundamental dimensions are the same as that for energy or work, official SI literature suggests using the unit newton metre (Nm) and never the joule.

Calculate the angle between a two dyne and a three dyne force so that their sum is four dynes.

The velocity of 20m/s has it's x-component 12 m/s. What is its y component? Find the angle at which the velocity is inclined with the x-axes.

The y component is a number such that

20^2 + y^2

= 20^2 or y^2

= 400 - 144

= 256

= 16^2 so the y component is 16 m/s and the angle is acos(12/20) = acos(3/5)

= acos(0.6) = approx 53.1 degrees.

Numerical Physics

Is pressure a vector?

pressure is a scalar quantity,not a vector quantity. It has a magnitude but no direction associated with it. Pressure acts in all directions at a point inside a gas. At the surface of a gas, the pressure force acts perpendicular to the surface.

Can the sum of three non-zero vectors give zero resultant?

If vectors are arranged as trigonal planar and have equal magnitude, it is an ideal case of zero resultant.