Question

if (a vector + b vector) perpendicular to b vector and (a vector+2b vector) perpendicular to a vector, then
prove a =√2 b​

Answer 1

Answer:

It's actually not true that  a = √2 b (unless a = b = 0),...

But it is true that |a| = √2 |b|.  Presumably this is what you meant!

Let's see...

    a+b perpendicular to b

⇒ (a + b)·b = 0

⇒  a·b + b·b = 0      ...(1)

    a+2b perpendicular to a

⇒ (a+2b)·a = 0

⇒  a·a + 2a·b = 0      ...(2)

Subtracting -2 times (1) from equation (2) gives

    a·a - 2b·b = 0   ⇒   a.a = 2b.b   ⇒   |a|² = 2 |b|²   ⇒   |a| = √2 |b|.

To see that it is not true that  a = √2 b when a and b are not the zero vector, it is enough to see that a and b do not have the same direction.

If a and b had the same direction, then a+b would have the same direction, too.  In particular, a+b would not be perpendicular to b.

Answer 2

hi!!!! the final answer is 1/root2