Question

In a parallelogram ABCD, diagonal vectors are `bar"AC" = 2hat"i" + 3hat"j" + 4hat"k" and bar"BD" = - 6hat"i" + 7hat"j" - 2hat"k"`, then find the adjacent side vectors `bar"AB" and bar"AD"`.

Answer 1

ABCD is a parallelogram.

∴ `bar"AB" = bar"DC", bar"AD" = bar"BC"`

`bar"AC" = bar"AB" + bar"BC"`

`= bar"AB" + bar"AD"`     ...(1)

`bar"BD" = bar"BA" + bar"AD" = - bar"AB" + bar"AD"`    ...(2)

Adding (1) and (2), we get

`2bar"AD" = bar"AC" + bar"BD" = (2hat"i" + 3hat"j" + 4hat"k") + (- 6hat"i" + 7hat"j" - 2hat"k")`

`= - 4hat"i" + 10hat"j" + 2hat"k"`

∴ `bar"AD" = 1/2(- 4hat"i" + 10hat"j" + 2hat"k")`

`= - 2hat"i" + 5hat"j" + hat"k"`

From (1), `bar"AB" = bar"AC" - bar"AD"`

`= (2hat"i" + 3hat"j" + 4hat"k") - (- 2hat"i" + 5hat"j" + hat"k")`

`= 4hat"i" - 2hat"j" + 3hat"k"`