If ` vec a , a n d vec b` are unit vectors , then find the greatest value of `| vec a+ vec b|+| vec a- vec b|dot`

Options

A. 2
B. 4
C. `2sqrt2`
D. `sqrt2`

Answer

C

Explanation

Let `theta` be an angle between unit vectors a and b. Then, `a.b =cos theta`
Now, `|a+b|^(2)=|a|^(2)+|b|^(2)+2a.b`
`=1+1+2xx1xx1xxcos theta`
`=2+2 cos theta=4 "cos"^(2) theta/2`
`implies |a+b|=2 "cos" theta/2`
and `|a-b|^(2)=|a|^(2)+|b|^(2)-2a.b`
`=2-2 cos theta=2 (2 "sin"^(2) theta/2)`
`|a-b|=2"sin" theta/2`
`:. |a+b|+|a-b|=2 ("cos" theta/2 +"sin"theta/2) le 2 sqrt(2)`