What are the maximum and minimum value of A → · B →

As A . → B → = A B cos θ Maximum and minimum values depend on angle between the two vectors W h e n cos θ = 0 = m i n ⇒ θ = 90 ∘ W h e n cos θ = 1 = m a x ⇒ θ = 0 ∘

What is the angle between A → and m A → where m is a constant ?

Let's take vector a → = a 1 i ^ + a 2 j ^ + a 3 k ^ Now multiply vector a → with scalar m m a → = m a 1 i ^ + a 2 j ^ + a 3 k ^ So, angle between two vectors is given by cos θ = a → . m a → a → m a → cos θ = m a 1 2 + m a 2 2 + m a 3 2 a 1 2 + a 2 2 + a 3 2 ( m a 1 2 + m a 2 2 + m a 3 2 ) cos θ = m a 1 2 + a 2 2 + a 3 2 a 1 2 + a 2 2 + a 3 2 m 2 ( a 1 2 + a 2 2 + a 3 2 ) cos θ = m a 1 2 + a 2 2 + a 3 2 a 1 2 + a 2 2 + a 3 2 . m ( a 1 2 + a 2 2 + a 3 2 ) cos θ = a 1 2 + a 2 2 + a 3 2 a 1 2 + a 2 2 + a 3 2 a 1 2 + a 2 2 + a 3 2 cos θ = a 1 2 + a 2 2 + a 3 2 a 1 2 + a 2 2 + a 3 2 cos θ = 1 ⇒ θ = 0 ∘ Therefore, the angle between the two given vectors is zero degree.