Question

Two sides of a parallelogram are `3hat"i" + 4hat"j" - 5hat"k"` and  `-2hat"j" + 7hat"k"`. Find unit vectors parallel to the diagonals.

Answer 1

Let ABCD be a parallelogram with

`bar"AB" = 3hat"i" + 4hat"j" - 5hat"k" and bar"BC" = - 2hat"i" + 7hat"k"`

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

`= (3hat"i" + 4hat"j" - 5hat"k") + (- 2hat"i" + 7hat"k")`

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

∴ `|bar"AC"| = sqrt(3^2 + 2^2 + 2^2) =sqrt(9 + 4 + 4) = sqrt17`

∴ unit vector along `bar"AC" = bar"AC"/|bar"AC"|`

= `1/sqrt17 (3hat"i" + 2hat"j" + 2hat"k")`

Also, `bar"BD" = bar"BA" + bar"AD" = - bar"AB" + bar"BC" = bar"BC" - bar"AB"`

`= (- 2hat"i" + 7hat"k") - (3hat"i" + 4hat"j" - 5hat"k")`

`= - 3hat"i" - 6hat"j" + 12hat"k"`

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

∴ `|bar"BD"| = 3sqrt((-1)^2 + (-2)^2 + 4^2) = 3sqrt(1 + 4 + 16) = 3sqrt21`

∴ unit vector along `bar"BD" = bar"BD"/|bar"BD"|`

`= (3(- hat"i" - 2hat"j" + 4hat"k"))/(3sqrt21)`

`= 1/sqrt21 (- hat"i" - 2hat"j" + 4hat"k")`

Hence, the unit vectors parallel to the diagonals are

`1/sqrt17 (3hat"i" + 2hat"j" + 2hat"k") and 1/sqrt21 (- hat"i" - 2hat"j" + 4hat"k")`