Question

If the vector `vecb = 3hatj + 4hatk` is written as the sum of a vector `vec(b_1)`, parallel to `veca = hati + hatj` and a vector `vec(b_2)`, perpendicular to `veca`, then `vec(b_1) xx vec(b_2)` is equal to ______.

पर्याय

• `-3hati + 3hatj - 9hatk`

• `6hati - 6hatj + 9/2hatk`

• `-6hati + 6hatj - 9/2hatk`

• `3hati - 3hatj + 9hatk`

Answer 1

If the vector `vecb = 3hatj + 4hatk` is written as the sum of a vector `vec(b_1)`, parallel to `veca = hati + hatj` and a vector `vec(b_2)`, perpendicular to `veca`, then `vec(b_1) xx vec(b_2)` is equal to `underlinebb(6hati - 6hatj + 9/2hatk)`.

Explanation:

Given, `vecb = 3hatj + 4hatk`

According to the question,

`vecb = vecb_1 + vecb_2`  ...(i)

Here, `b_1 || veca` and `b_2 ⊥ a`

`\implies b_1 = λveca`

and b₂ = `b_2.veca` = 0

now b₁ = `λ(hati + hatj)`  ...`{∵ hata = hati + hatj}`

from (1) `vecb_2 = vecb - vecb_1`

So, `vecb_2.veca` = 0

`\implies (vecb - vecb_1)(hati + hatj)` = 0

`\implies vecb(hati - hatj) - λhata(hati - hatj)` = 0

`\implies (3hatj + 4hatk)(hati - hatj) - λ(hati - hatj)(hati - hatj)` = 0

`\implies` λ = `3/2`

now `vecb_2 = vecb - vecb_1`

`\implies (3hatj + 4hatk) - 3/2(hati + hatj)`

`\implies (-3)/2hati + 3/2hatj + 4hatk`

and `vec(b_1) xx vec(b_2) = |(hati, hatj, hatk),(3/2, 3/2, 0),(-3/2, 3/2, 4)|`

`\implies vec(b_1) xx vec(b_2) = hati(6) - hatj(6) + hatk(-9/4 + 9/4)`

`\implies 6hati - 6hatj + 9/2hatk`