Let ` vec u` be a vector on rectangular coordinate system with sloping angle `60^0dot` Suppose that `| vec u- hat i|` is geometric mean of `| vec u|a n d| vec u-2 hat i|,w h e r e hat i` is the unit vector along the x-axis. Then find the value of `(sqrt(2)+1)| vec u|dot`

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`