Let `vecu` be a vector on rectangular coodinate system with sloping angle `60^(@)` suppose that `|vecu-hati|` is geomtric mean of `|vecu| and |vecu-2hati|`, where `hati` is the unit vector along the x-axis . Then find the value of `(sqrt2- 1)/ |vecu|`

Answer

1

Explanation

since angle between `vecu and hati is 60^(@)` we have
`vecu. I = |vecu||hati|cos 60^(@) = (|vecu|)/2`
Given that ` |vecu - hati| ,|vecu| , |vecu -2hati|` are in G.P. so
`|vecu - hati|^(2)= |vecu| |vecu -2 hati|`
squaring both sides,
`[|vecu|^(2)+|hati|^(2)-2vecu.hati]^(2)=|vecu|^(2)[|vecu|^(2)+4|hati|^(2)-4vecu.hati]`
`[|vecu|^(2)+1-(2|vecu|)/2]^(2)=|vecu|^(2)[|vecu|^(2)+4-4(|vecu|)/2]`
`or |vecu|^(2)+ 2|vecu|-1=0Rightarrow|vecu|=-(2+-2sqrt2)/2`
`or |vecu|= sqrt2-1`