Given a parallelogram `ABCD`. If `|vec(AB)|=a, |vec(AD)| = b & |vec(AC)| = c` , then ` vec(DB) . vec(AB)` has the value

Options

A. `(1)/(2)(a^(2)+b^(2)+c^(2))`
B. `(1)/(2)(a^(2)-b^(2)+c^(2))`
C. `(1)/(4)(a^(2)+b^(2)-c^(2))`
D. `(1)/(3)(b^(2)+c^(2)-a^(2))`

Answer

A

Explanation

We have, `|AB|=a, |AD|=b, |AC|=c`

In `Delta ABC, " " AB + BC = AC `
`rArr AB+AD=AC " " [ because BC=AD]`
`rArr AB-DA=AC " " [ because AD=-DA]`
`rArr |AB-DA|=|AC|`
`rArr |AB-DA|^(2)=|AC|^(2)`
`rArr |AB|^(2)+|DA|^(2)-2AB*DA=(AC)^(2)`
`rArr a^(2)+b^(2)-2AB*DA=c^(2) " " [ because |AD|=|DA|=b]`
`rArr 2DA *AB=a^(2)+b^(2)-c^(2)`
`rArr DA*AB=(1)/(2)[a^(2)+b^(2)-c^(2)]`