Question

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

Options

• `sqrt(2)`

• `sqrt(3)`

• 1

• 2

Answer 1

If the sum of two unit vectors is itself a unit vector, then the magnitude of their difference is `underline sqrt(3)`.

Explanation:

Step 1: Given data

Let `vec a` and `vec b` be two unit vectors whose sum is also a unit vector `vec c`.

`vec a + vec b = vec c`

Also, `|vec a| = |vec b| = |vec c| = 1`

Step 2: Calculating the magnitude of the sum of the two vectors

Now, the magnitude of the sum of `vec a and vec b` is

`|vec a + vec b|^2 = |vec a|^2 + |vec b|^2 + 2 ("ab cos" theta)`

1 = 1 + 1 + 2(ab cos θ)

(ab cos θ) = `-1/2`

Step 3: Calculating the magnitude of the difference between the two vectors

The magnitude of their difference is given by

`|vec a - vec b|^2 = |vec a|^2 + |vec b|^2 - 2("ab cos" theta)`

`1 + 1 - 2 xx (-1/2)` = 3

`|vec a - vec b|^2 = 3`          ...[Square root on both sides]

`|vec a - vec b| = sqrt3`

Therefore, the magnitude of the difference between the two vectors is `sqrt3`