Vectors `3veca-5vecb and 2veca + vecb` are mutually perpendicular. If `veca + 4 vecb and vecb - veca` are also mutually perpendicular, then the cosine of the angle between `veca` and `vecb` is (a) `19/(5sqrt43)` (b) `19/(3sqrt43)` (c) `19/(sqrt45)` (d) `19/(6sqrt43)`

Options

A. `19/(5sqrt43)`
B. `19/(3sqrt43)`
C. `19/(sqrt45)`
D. `19/(6sqrt43)`

Answer

a

Explanation

`(3veca-5vecb).(2veca+vecb)=0`
`or 6|veca|^(2)-5|vecb|^(2)= 7 veca.vecb`
`Also , (veca+4vecb).(vecb-veca)=0`
`or -|veca|^(2)+ 4|vecb|^(2)= 3 veca.vecb`
`or 6/7 |veca|^(2)-5/7|vecb|^(2)= -1/3 |veca|^(2)+ 4/3|vecb|^(2)`
` or 25|veca|^(2) = 43 |vecb|^(2)`
` Rightarrow 3 veca. vecb= - |veca|^92) + 4|vecb|^(2)= 57/25 |vecb|^(2)`
`or 3 |veca||vecb| cos theta = 57/25 |vecb|^(2)`
`or 3sqrt(43/25) |vecb|^(2) cos theta = 57/25 |vecb|^(2)`
` or cos theta = 19/(5sqrt43)`