Question

Vectors `veca = 3hati - 2hatj + 2hatk and vecb = hati + 2hatk` represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.

Answer 1

Given:

`veca = 3hati - 2hatj + 2hatk and vecb = hati + 2hatk`

Diagonals of a parallelogram:

⇒ `vecd_1 = veca + vecb`

= `(3 + 1)hati + (-2)hatj + (2 + 2)hatk`

∴ `vecd_1 = 4hati - 2hatj + 4hatk`

⇒ `vecd_2 = veca - vecb`

= `(3 - 1)hati - 2hatj + (2 - 2)hatk`

∴ `vecd_2 = 2hati - 2hatj`

Lengths:

⇒ `|vecd_1| = sqrt(4^2 + (-2)^2 + 4^2)`

= `sqrt(16 + 4 + 16)`

= `sqrt36`

∴ `|vecd_1| = 6`

⇒ `|vecd_2| = sqrt(2^2 + (-2)^2 + 0)`

= `sqrt(4 + 4)`

= `sqrt8`

= `2sqrt2`

∴ `|vecd_2| = 2sqrt2`

Hence, the vectors representing its diagonals and their lengths are:

`vecd_1 = 4hati - 2hatj + 4hatk, |vecd_1| = 6`

`vecd_2 = 2hati - 2hatj, |vecd_2| = 2sqrt2`