Question

When will the sum and difference of two vectors be equal in magnitude?

Answer 1

Logically, how can magnitude of vector sum be equal to difference of their magnitudes. Obviously if the vectors are in opposite directions. This means the angle has to be 180.

Mathematically,

Let the vectors be a and b with magnitudes a and b respectively and the angle between them be x.

Magnitude of the sum of a and b is

√(a^2+b^2+2abcosx

Difference in their magnitudes is

a-b

Hence,

√(a^2+ b^2+2ab cosx) = a-b

Squaring both sides,

a^2+b^2+2ab cos x = a^2+ b^2–2ab

2ab cosx+2ab =0

2ab(cosx +1) =0

Since 2ab can't be zero,

Cos x+1=0

Cosx=-1

X=180