State and prove parallelogram law of force Elements of Civil Engineering and Mechanics

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State and prove parallelogram law of force.-Elements of Civil Engineering and Mechanics

Elements of Civil Engineering and Mechanics


By:milan-ransingh

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|prajwalamv

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The Statement of Parallelogram law of vector addition is,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.

Let a,ba,b be two vectors & let these constitute the sides of a parallelogram such that they are co-initial from one of the vertices of a parallelogram. So, considering assumption that the law be true(this is what we'll judge now whether by taking the assumption for the validity of the law, we can prove something which is earlier ascertained by elementary geometry), the two diagonals are a+b,b−aa+b,b−a. Sum of the squares of the diagonals is |a+b|2+|b−a|2|a+b|2+|b−a|2. Now, from Euclidean Geometry, we get from Parallelogram law which states that,

The sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals.11
So, the sum of the squares of the diagonals i.e. |a+b|2+|b−a|2|a+b|2+|b−a|2 must be equal to the sum of the squares of the sides 2(|a|^2+|b|^2)2(|a|^2+|b|^2).

This can be proved by inner product or for this case dot product.

|a+b|^2+|b−a|^2=(a+b)⋅(a+b)+(b−a)⋅(b−a)⟹|a+b|^2+|b−a|^2=2a⋅a+2b⋅b=2(|a|^2+|b^|2)
|a+b|^2+|b−a|^2=(a+b)⋅(a+b)+(b−a)⋅(b−a)⟹|a+b|^2+|b−a|^2=2a⋅a+2b⋅b=2(|a^|2+|b|^2)
which makes our assumption that the law is true absolutely true.

prajwalamv

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