can sum of magnitude of two vector be equal to magnitude of sum of the same vectors

Dear Student,

 

The sum of magnitude of two vector can be equal to magnitude of sum of the same vectors provided the two vectors have the same direction.

 

For eg. If a force 4N is applied over a box towards east and also a force of 5N is aaplied over the same box towards east then the net force over the box becomes to be 9N.

 

We can prove this as given below -

We know that [vector(a) + vector(b)]^2 = [a^2 +b^2 + 2abcosθ]

therefore for same direction, θ = 0 , ie, cosθ = 1

thus

[vector(a) + vector(b)]^2 = [a^2 +b^2 + 2ab(1)]

[vector(a) + vector(b)]^2 = [a^2 +b^2 + 2ab]

[vector(a) + vector(b)]^2 = [a +b]^2

 

hence, [vector(a) + vector(b)] = [a+b] (for θ = 0)