Question

If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is `sqrt(3)`.

Answer 1

Let three unit vectors are a, b and c
given that the sum of the unit vectors is a unit vector.
∴ a + b = c

or  | c |² = | a + b |²

or | c |² = | a |² + | b |² + 2| a | | b |cos θ

or 1 = 1 + 1 + 2 cos θ   ...[∵ | a | = | b | = | c | = 1 (unit vector)]

⇒ `cos θ = -1/2`   ...(1)

Now, | a - b |² =| a |² + | b |² - 2| a | | b |cos θ

| a - b |² = [ 1 + 1 + 1 ]

`|a -b|= sqrt(3)`