Question

If `veca = hati + 2hatj + 3hatk` and `vecb = 2hati + 4hatj - 5hatk` represent two adjacent sides of a parallelogram, find unit vectors parallel to the diagonals of the parallelogram.

Answer 1

Given that, `veca = hati + 2hatj + 3hatk` and `vecb = 2hati + 4hatj - 5hatk` are two adjacent sides of a parallelogram.

Let us suppose `vec(d_1)` and `vec(d_2)` are two diagonals of parallelogram.

Then, `vec(d_1) = veca + vecb`

= `hati + 2hatj + 3hatk + 2hati + 4hatj - 5hatk`

= `3hati + 6hatj - 2hatk`

And `vec(d_2) = vecb - veca`

= `2hati + 4hatj - 5hatk - hati - 2hatj - 3hatk`

= `hati + 2hatj - 8hatk`

Now, unit vector parallel to `vec(d_1)` is

`hat(d)_1 = (3hati + 6hatj - 2hatk)/sqrt(9 + 36 + 4)`

= `(3hati + 6hatj - 2hatk)/sqrt(49)`

`hat(d)_1 = (3hati + 6hatj - 2hatk)/7`

And unit vector parallel to `vec(d_2)` is

`vec(d_2) = (hati + 2hatj - 8hatk)/sqrt(1 + 4 + 64)`

= `(hati + 2hatj - 8hatk)/sqrt(69)`