β + γ Example Find the angle between two vectors with magnitudes and respectively and when = . Solution Given . We have Example Find angle ‘ θ ’ between the vectors Solution The angle θ between two vectors is given by cos θ = Now ) ( = −−= − . Therefore, we have cos θ = hence the required angle is θ = Example If , then show that the vectors are perpendicular.
Solution We know that two nonzero vectors are perpendicular if their scalar product is zero. Here ( ) ) ( ) ) So ) . ( ( ) ( ) . Hence are perpendicular vectors.
Example Find the projection of the vector on the vector Solution The projection of vector on the vector b is given by ( ) ( ) ( ) ( ) × + × + × Example Find , if two vectors are such that Solution We have B C A ( ) ( ) ( ) Therefore Example If is a unit vector and , then find Solution Since is a unit vector, . Also, or = or Therefore = (as magnitude of a vector is non negative). Example For any two vectors , we always have (Cauchy- Schwartz inequality). Solution The inequality holds trivially when either or r a .
Actually, in such a situation we have . So, let us assume that Then, we have = | cos | θ ≤ Therefore Example For any two vectors , we always have (triangle inequality). Solution The inequality holds trivially in case either (How?). So, let .
Then, (scalar product is commutative) ≤ (since | | x x x ≤ ∀∈ R ) ≤ (from Example ) Fig . Hence Remark If the equality holds in triangle inequality (in the above Example ), i.e. then showing that the points A, B and C are collinear. Example Show that the points A(