Find the angle between the vectors a → = i ^ + j ^ - k ^ and b → = i ^ - j ^ + k ^

The formula to find the angle between two vectors is given by cos θ = a → · b → a → b → Let a → = i ^ + j ^ - k ^ and b → = i ^ - j ^ + k ^ . Now, a → · b → = ( i ^ + j ^ - k ^ ) · ( i ^ - j ^ + k ^ ) = ( 1 × 1 ) + ( 1 × - 1 ) + ( - 1 × 1 ) = - 1 The magnitude of the vectors is a → = 1 2 + 1 2 + ( - 1 ) 2 = 1 + 1 + 1 = 3 and b → = 1 2 + ( - 1 ) 2 + 1 2 = 1 + 1 + 1 = 3 Substituting the values in the equation, cos θ = - 1 3 × 3 = - 1 3 ⇒ θ = cos - 1 - 1 3

Find a → - b → , if two vector a → and b → are such that a → = 2 , b → = 3 and a → · b → = 4 .

Given that, We have to find a → - b → , if two vectors a → and b → are such that a → = 2 , b → = 3 and a → . b → = 4 . Consider a → - b → 2 = a → - b → 2 = a → - b → . a → - b → . = a → . a → - a → . b → - b → . a → + b → . b → = a → 2 - a → . b → - a → . b → + b → 2 ∵ a → . b → = b → . a → = a → 2 - 2 a → . b → + b → 2 ⇒ a → - b → 2 = 2 2 - 2 × 4 + 3 2 = 5 ⇒ a → - b → = 5 We have considered positive root because magnitude is always positive. Therefore, the value of a → - b → is 5 .