Question

If `overlinea = hati + hatj + hatk` and `overlinec = hatj - hatk` and `overlineb` is a vector satisfying `overlinea xx overlineb = overlinec` and `overlinea . overlineb = 3`, then `3|overlineb|^2` is equal to ______ 

Options

• 11

• 22

• 33

• 44

Answer 1

If `overlinea = hati + hatj + hatk` and `overlinec = hatj - hatk` and `overlineb` is a vector satisfying `overlinea xx overlineb = overlinec` and `overlinea . overlineb = 3`, then `3|overlineb|^2` is equal to 11.

Explanation:

Let `overlineb = b_1hati + b_2hatj + b_3hatk`

`overlinea . overlineb = 3`

⇒ `(hati + hatj + hatk) . (b_1hati + b_2hatj + b_3hatk) = 3`

⇒ `b_1 + b_2 + b_3 = 3` .....................(i)

`overlinea xx overlineb = overlinec`

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

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

⇒ `b_3 - b_2 = 0, b_1 - b_3 = 1, b_1 - b_2 = 1` ...................(ii)

From (i) and (ii), we get

`b_1 = 5/3, b_2 = 2/3, b_3 = 2/3`

⇒ `overlineb = 5/3hati + 2/3hatj + 2/3hatk`

⇒ `|overlineb|^2 = 11/3`

⇒ `3|b|^2 = 11`