Question

A state law of parallelogram law of
Vectors perive an expression for the
magnitude and direction of the
resultant vector.​

Answer 1

Answer:

Parallelogram law states that if two vectors are considered to be the adjacent sides of a Parallelogram, then the resultant of two vectors is given by the vector which is a diagonal passing through the point of contact of two vectors.

If ABCD is a parallelogram, then AB = DC and AD = BC. Then according to the definition of the parallelogram law, it is stated as

2(AB)2 + 2 (BC)2 = (AC)2 + (BD)2.

In case the parallelogram is a rectangle, then the law is stated as:

2(AB)2 + 2 (BC)2 = 2(AC)2

Because in rectangle, two diagonals are of equal lengths. i.e., (AC = BD)

Parallelogram Law Proof

Let AD=BC = x, AB = DC = y, and ∠ BAD = α

Using the law of cosines in the triangle BAD, we get

x2 + y2 – 2xy cos(α) = BD2 ——-(1)

We know that in a parallelogram, the adjacent angles are supplementary. So

∠ADC = 180 – α

Now, again use the law of cosines in the triangle ADC

x2 + y2 – 2xy cos(180 – α) = AC2 ——-(2)

Apply trigonometric identity cos(180 – x) = – cos x in (2)

x2 + y2 + 2xy cos(α) = AC2

Now, the sum of the squares of the diagonals (BD2 + AC2) are represented as,

BD2 + AC2  = x2 + y2 – 2xycos(α) + x2 + y2 + 2xy cos(α)

Simplify the above expression, we get;

BD2 + AC2 =2x2 + 2 y2 ——-(3)

The above equation is represented as:

BD2 + AC2 = 2(AB)2 + 2(BC)2