Question

By keeping the direction of a vector the same if its magnitude is doubled, would the magnitude of its every component be doubled?

Answer 1

Explanation:

Let’s say we have two vectors initially , a & b with their resultant is r .

When magnitude of one vector (say b) gets doubled the vectors became a & c with their resultant R .

we know that, |c| = 2 |b| also a ⊥ R → a.R = 0

We only know one formula to calculate the magnitude of resultant so lets apply that,

|r| = √(a^2 + b^2 + 2*(a.b) ) ……(1)

We know this, a.R = 0

as, R = a + c

R = a + 2b

….(we replaced c with 2b because |c| = 2 |b| and they have same direction)

So, a.(a + 2b ) = 0

a.a + 2*(a.b) =0

a^2 + 2*(a.b) =0 …….(2)

From 1 and 2 ,

|r| = √( b^2 )

|r| = b

so magnitude of previous resultant r is equal to the vector which gets doubled..

The magnitude of new resultant R can’t be found out until & unless the angle and more data is given…