devotion & mathematics

For a video explanation , click the link   https://youtu.be/YB_p6iPSIfQ    Theorem :             Let V(F) be a vector space and 0 and  O be the zero scalar and zero vector respectively. Then    (i)  a O = O ∀ a 𝞊 F.   (ii) o 𝜶 = O ∀ a 𝞊 F, 𝜶 𝞊 V     (iii) a ( - 𝜶 ) = - (a𝜶) = (-a) 𝜶 ∀ a 𝞊 F , 𝜶 𝞊 V   (iv) a ( 𝜶 - 𝛃 ) = a𝜶 - a𝛃 ∀ a 𝞊 F , 𝜶,𝛃 𝞊 V.    Proof :                            * External Composition   *  Historical Introduction to Linear Algebra  * What is a vector space *   Let V(F) be a vector space and let W ⊆ V. The necessary and sufficient conditions for W   to be a subspace of V are (i) 𝜶 𝞊 W, 𝞫 𝞊 W ⇒  𝜶 - 𝞫 𝞊 W     (ii) a 𝞊 F , 𝜶 𝞊 W ⇒ a𝜶 𝞊 W. #vector #algebra #vector algebra   #iit #jee #mains #dimens...

***Few problems & Solutions on derivatives of a vector functions *** Problem 1 :               If r = a cost i + a sint j + at tanθ k then find (i) |(dr/dt)  x (d 2 r/dt 2 ) |                                                                                    (ii)  [ (dr/dt)    (d 2 r/dt 2 )   (d 3 r/dt 3 )  ] Solution : Problem 2 :      If r= e t (c cos2t+ d sin2t) where c and d are constant vectors, then show that d 2 r/dt 2 -2 dr/dt  +5r = 0. Solution :  Problem 3 :            If r = t 2  i -t j + (2t+1) k, find the values of dr/dt ,d 2 r/dt 2   , |dr/dt| and | d 2 r/dt 2 | at t=0. Solution :           ...