Question

two vectors A and B and added prove that the magnitude of the resultant vectors cannot be greater than (A+B)or(A-B)​

Answer 1

Given :-

Two vectors A and B are added

To prove :-

Magnitude of Resultant of two vectors cannot be greater than ( A + B ) and smaller than ( A - B )

Knowledge required :-

The magnitude of resultant of two vectors M and N is given by

▶        R² = M² + N² + 2 MN cos θ

( where R is the magnitude of resultant of two vectors ,and M and N are magnitude of two vectors M and N ; θ is the angle between vectors M and N )

Solution :-

Let, magnitude of resultant of vectors A and B be R

We know that ,

greatest value of cos θ is 1 at 0°

and

smallest value of cos θ is -1 at 180°

so,

Taking θ = 0°

→  R² = A² + B² + 2 AB cos 0°

→ R² = A² + B² + 2 AB ( 1 )

→ R² = A² + B² + 2 AB

→ R² = ( A + B )²

→ R = ( A + B )

Taking θ = 180°

→ R² = A² + B² + 2 AB cos 180°

→ R² = A² + B² + 2 AB ( - 1 )

→ R² = A² + B² - 2 AB

→ R² = ( A - B )²

→ R = ( A - B )

Therefore,

Magnitude of resultant of two vectors cannot be greater than ( A + B ) and smaller than ( A - B ).