Question

If the sum of two unit vectors is a unit vector, then magnitude of difference is ______.

Options

• \[\sqrt 2\]

• \[\sqrt 3\]

• \[\sqrt \frac {1}{2}\]

• \[\sqrt 5\]

Answer 1

If the sum of two unit vectors is a unit vector, then magnitude of difference is \[\sqrt 3\].

Explanation:

Let \[\hat {n_1}\] and \[\hat {n_2}\] be the two unit vectors, then the sum is

\[\begin{array} {rcl}\mathrm{n_s} & = & \hat{\mathrm{n_1}} & + & \hat{\mathrm{n_2}} \end{array}\]

\[\begin{array} {rcl}\mathrm{n_s^2~=~n_1^2~+~n_2^2~+2n_1n_2~\cos\theta=1~+~1~+2~\cos\theta} \end{array}\]

As n_s is also a unit vector,

⇒ 1 = 1 + 1 + 2 cos θ

\[\therefore\] cos θ = -\[\frac {1}{2}\] ⇒ θ = 120°

Let the difference vector be \[\hat n_d\] = \[\hat n_1\] - \[\hat n_2\]

n_d² = n₁² + n₂² - 2n₁n₂ cos θ

= 1 + 1 - 2cos(120°)

\[\therefore\] n_d² = 2 - 2(-1/2) = 2 + 1 = 3

\[\therefore\] \[\sqrt 3\]