Question

Let `veca = 2hati + hatj - 2hatk, vecb = hati + hatj`. If `vecc` is a vector such that `veca . vecc = \|vecc|, |vecc - veca| = 2sqrt(2)` and the angle between `veca xx vecb` and `vecc` is 30°, then `|(veca xx vecb) xx vecc|` equals ______.

Options

• `1/2`

• `(3sqrt(3))/2`

• 3

• `3/2`

Answer 1

Let `veca = 2hati + hatj - 2hatk, vecb = hati + hatj`. If `vecc` is a vector such that `veca . vecc = \|vecc|, |vecc - veca| = 2sqrt(2)` and the angle between `veca xx vecb` and `vecc` is 30°, then `|(veca xx vecb) xx vecc|` equals `underlinebb(3/2)`.

Explanation:

`veca = 2hati + hatj - 2hatk, vecb = hati + hatj`

`\implies |veca|` = 3

and `veca xx vecb = |(hati, hatj, hatk),(2, 1, -2),(1, 1, 0)| = 2hati - 2hatj + hatk`

`|veca xx vecb| = sqrt(4 + 4 + 1)` = 3

Now, `|vecc - veca| = 2sqrt(2) \implies |vecc - veca|^2` = 8

`\implies |vecc - veca|.(vecc - veca)` = 8

`\implies |vecc|^2 + |veca|^2 - 2vecc.veca` = 8

`\implies |vecc|^2 + 9 - 2|vecc|` = 8

`\implies (|vecc| - 1)^2` = 0

`\implies |vecc|` = 1

∴ `|(veca xx vecb) xx vecc| = |veca xx vecb||vecc| sin 30^circ = 3 xx 1 xx 1/2 = 3/2`