Question

Let a vector `veca` be coplanar with vectors `vecb = 2hati + hatj + hatk` and `vecc = hati - hatj + hatk`. If `veca` is perpendicular to `vecd = 3hati + 2hatj + 6hatk` and `|veca| = sqrt(10)`. Then a possible value of `[(veca, vecb, vecc)] + [(veca, vecb, vecd)] + [(veca, vecc, vecd)]` is equal to ______.

Options

• –42

• –40

• –29

• –38

Answer 1

Let a vector `veca` be coplanar with vectors `vecb = 2hati + hatj + hatk` and `vecc = hati - hatj + hatk`. If `veca` is perpendicular to `vecd = 3hati + 2hatj + 6hatk` and `|veca| = sqrt(10)`. Then a possible value of `[(veca, vecb, vecc)] + [(veca, vecb, vecd)] + [(veca, vecc, vecd)]` is equal to –42.

Explanation:

Let `veca = xhati + yhatj + zhatk`

Since `veca, vecb` and `vecc` are coplanar

∴ `[(veca, vecb, vecc)] = |(x, y, z),(2, 1, 1),(1, -1, 1)|` = 0

`\implies` 2x – y – 3z = 0  ...(i)

and `veca . vecd` = 0

3x + 2y + 6z = 0  ...(ii)

From (i) and (ii)

`x/|(-1, -3),(2, 6)| = (-y)/|(2, -3),(3, 6)| = z/|(2, -1),(3, 2)|` = λ

`\implies` x = 0, y = –21λ, z = 7λ

Given x² + y² + z² = 10

441λ² + 49λ² = 10

λ² = `1/49`

`\implies` λ = `±1/7`

∴ `veca = -3hatj + hatk` or `veca = 3hatj - hatk`

Now, `[(veca, vecb, vecc)] + [(veca, vecb, vecd)] + [(veca, vecc, vecd)] = [vecavecb + veccvecd]`

= `|(0, -3, 1),(3, 0, 2),(3, 2, 6)|` or `|(0, 3, -1),(3, 0, 2),(3, 2, 6)|`

= 42 or –42