Question

If `veca = hati + hatj + hatk, veca.vecb` = 1 and `veca xx vecb = hatj - hatk`, then find `|vecb|`.

Answer 1

Given, `veca = hati + hatj + hatk`

`veca.vecb` = 1

And `veca xx vecb = hatj - hatk`

Let `vecb = ahati + bhatj + chatk`

Now, `veca.vecb` = 1

⇒ `(hati + hatj + hatk)(b_1hati + b_2hatj + b_3hatk)` = 1

⇒ `b_1 + b_2 + b_3` = 1  ...(i)

And `veca xx vecb = hatj - hatk`

⇒ `|(hati, hatj, hatk),(1, 1, 1),(b_1, b_2, b_3)| = hatj - hatk`

⇒ `hati(b_3 - b_2) - hatj(b_3 - b_1) + hatk(b_2 - b_1) = hatj - hatk`

On comparing both sides, we get

–(b₃ – b₁) = 1 and b₂ – b₁ = –1

⇒ b₃ – b₁ = –1 and b₂ – b₁ = –1

⇒ b₃ = –1 + a and b₂ = –1 + b₁  ...(ii)

Now from equation (i), we get

b₁ + (–1 + b₁) + (–1 + b₁) = 1

⇒ 3b₁ = 3

⇒ b₁ = 1

From equation (ii), we get

b₂ = 0 and b₃ = 0

∴ `vecb = hati`

Therefore, `|vecb|` = 1