Question

If the sum of two unit vectors is a unit vector, what is the magnitude of the difference of the two vectors?

Answer 1

Unit vector: The vector whose magnitude is unity(1) is called unit vector.

Explanation:

Let ai+bj and ci+dj be the unit vectors.
From the defination,
`sqrt(a^2+b^2)=1`
`=>(a^2+b^2)=1`__(1)
`sqrt(c^2+d^2)=1`
`=>(c^2+d^2)=1`_____(2)

Add the above two vectors,
(ai+bj)+(ci+dj) =(a+c)i+(b+d)j__(3)

Also given that,
The magnitude of equation(3) is also unity.
`sqrt((a+c)^2+(b+d)^2)=1`
`=>((a+c)^2+(b+d)^2)=1`
`=>(a^2+c^2+2ac)+(b^2+d^2+2bd)=1`
`=>1+2ac+1+2bd=1`
`=>cancel(1)+2ac+2bd+1=cancel(1)`
`=>ac+bd=-1/2`___(4)
the magnitude of their difference is:?
THe difference of two unit vectors is
`(ai+bj)-(ci+dj)=(a-c)i+(b-d)j`
`Magnitude=sqrt((a-c)^2+(b-d)^2`
`=sqrt(a^2+c^2-2ac+b^2+d^2-2bd)`
`=sqrt(1+1-2(ac+bd)` [From (1) and (2)]
`=sqrt(2-2(-1/2)` [From equation(4)]
`=sqrt(2+1)`
`=sqrt(3)`
`=1.73205`

If the sum of two unit vectors is a unit vector, then the magnitude of their difference is 1.73205

Answer 2

The magnitude (length) of the difference is `sqrt(3)`
(Alternate method of seeing this).

Explanation:

For those more visually or geometrically oriented; consider the following diagram, noting that if the sum of two unit vectors is a unit vector then the 3 unit vectors can be combined into an equilateral triangle:

By extending the initial unit vector and dropping a perpendicular to that extension from the terminal point of the vector `vec(a-b)`
we have a right triangle with sides:
`1+1/2` and `sqrt(3)/2`

So
`abs(vec(a-b)) = sqrt((3/2)^2 +(sqrt(3)/2)^2)`

`color(white)("XXXX")` `=sqrt((9+3)/4) = sqrt(12/4) = sqrt(3)`