Let v → = α i ^ + 2 j ^ - 3 k ^ , w → = 2 α i ^ + j ^ - k ^ , and u → be a vector such that u → = α > 0 . If the minimum value of the scalar triple product u → v → w → is - α 3401 , and u → . i ^ 2 = m n where m and n are coprime natural numbers, then m + n is equal to _____ .

Step-by-step solution

We have, u →   v →   w → = u → · v → × w → Now, min u →   v →   w → = - α 3401 ⇒ - u → v → × w → = - α 3401 ⇒ α v → × w → = α 3401 ⇒ v → × w → = 3401 ⇒ i ^ j ^ k ^ α 2 - 3 2 α 1 - 1 = 3401 ⇒ i ^ - 5 α j ^ - 3 α k ^ = 3401 ⇒ 1 + 25 α 2 + 9 α 2 = 3401 ⇒ 34 α 2 = 3400 ⇒ α 2