the component form ˆ+ a i a j a k b i b j b k , respectively, then (i) the sum (or resultant) of the vectors is given by b i b k (ii) the difference of the vector is given by b i b k (iii) the vectors are equal if and only if a = b , a = b and a = b (iv) the multiplication of vector by any scalar λ is given by λ = a i a k λ + λ + λ The addition of vectors and the multiplication of a vector by a scalar together give the following distributive laws: Let be any two vectors, and k and m be any scalars. Then (i) (ii) (iii) Remarks (i) One may observe that whatever be the value of λ , the vector λ is always collinear to the vector . In fact, two vectors are collinear if and only if there exists a nonzero scalar λ such that . If the vectors are given in the component form, i.e.
= a i a j a k , then the two vectors are collinear if and only if b i b j b k a i a j a k λ ⇔ b i b j b k a i a k λ + λ + λ ⇔ = λ = λ = λ ⇔ a = = λ (ii) If = a i a j a k , then a , a , a are also called direction ratios of . (iii) In case if it is given that l , m , n are direction cosines of a vector, then li mj nk (cos ) (cos ) (cos ) α β γ is the unit vector in the direction of that vector, where α , β and γ are the angles which the vector makes with x , y and z axes respectively. Example Find the values of x , y and z so that the vectors are equal. Solution