Question

Let three vectors `veca, vecb` and `vecc` be such that `vecc` is coplanar with `veca` and `vecb, vecc,` = 7 and `vecb` is perpendicular to `vecc` where `veca = -hati + hatj + hatk` and `vecb = 2hati + hatk`, then the value of `2|veca + vecb + vecc|^2` is ______.

Options

• 74

• 75

• 76

• 77

Answer 1

Let three vectors `veca, vecb` and `vecc` be such that `vecc` is coplanar with `veca` and `vecb, vecc,` = 7 and `vecb` is perpendicular to `vecc` where `veca = -hati + hatj + hatk` and `vecb = 2hati + hatk`, then the value of `2|veca + vecb + vecc|^2` is 75.

Explanation:

Given: `vecc` is coplanar with `veca xx vecb`

⇒ `vecc = xveca + yvecb`

∵ `vecb.vecc` = 0  ⇒ `x(veca.vecb) + y(vecb.vecb)` = 0 ⇒ –x + 5y = 0  ...(i)

`vecc.veca` = 7 ⇒ `x(veca.veca) + y(veca.vecb)` = 7 ⇒ 3x – y = 7

⇒ y = `1/2`, x = `5/2`

∵ `vecc = (5veca + vecb)/2`

⇒ `2(veca + vecb + vecc) = 2veca + 2vecb + 5veca + vecb = 7veca + 3vecb`

Squaring the above equation both sides,

⇒ `4(veca + vecb + vecc)^2 = 49(veca.veca) + 9(vecb.vecb) + 42(veca.vecb)`

= 49(3) + 9(5) – 42(1) = 150

⇒ `2(veca + vecb + vecc)^2` = 75