Vectors `3 vec a-5 vec b` and `2 vec a+ vec b` are mutually perpendicular. If `vec a+4 vec b` and `vec b- vec a` are also mutually perpendicular, then the cosine of the angle between `a and b` is a.`(19)/(5sqrt(43))` b. `(19)/(3sqrt(43))` c. `(19)/(2sqrt(45))` d. `(19)/(6sqrt(43))`

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)`