Title 3. VECTOR_SYN _ W.S-5 TO 7_VOL-1_(61 TO 90).pdf ✅

VIII – Physics (Vol – I) Olympiad Class Work Book Then  1 1 1 2 2 2    A B a i b j c k a i b j c k        ˆ ˆ ˆ ˆ ˆ ˆ   1 2 1 2 1 2 1 2 1 2                     a a i i a b i j a c i k b a j i b b j j ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 1 2 1 2 1 2 1 2                 b c j k c a k i c b k j c c k k ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Since, ˆ ˆ ˆ ˆ ˆ ˆ i i j j k k 0       and ˆ ˆ ˆ i j k   , etc., we have  1 2 1 2 1 2 1 2 1 2 1 2      A B b c c b i c a a c j a b b a k        ˆ ˆ ˆ   or putting it in determinant form, we have 1 1 1 2 2 2 ˆ ˆ ˆ i j k A B a b c a b c     It may be noted that the scalar components of the first vector A  occupy the middle row of the determinant. CROSS PRODUCT BY COMPONENTS METHOD (i) If the vectors are expressed in terms of their rectangular components along X, Y and Z axes, the cross product yields the product in vector form i.e. in terms of its rectangular components again. (ii) If A  = Ax i + A y j + Az k and B  = Bx i + B y j + Bz k where Ax , Ay and Az are the components of the vector A  along X , Y and Z axis. Similarly, Bx , By and Bz are the components of the vector B  along X, Y and Z axis respectively And i, j and k. are the unit vectors along X , Y and Z axis respectively. then A B    = (Ax i + A y j + Az k) × (Bx i + B y j + Bz k) = Ax i × (Bx i + B y j + Bz k) + Ay j × (Bx i + B y j + Bz k) + Az k × (Bx i + B y j + Bz k) = Ax i × Bx i + Ax i × B y j + Ax i × Bz k + A y j × Bx i + A y j × B y j + A y j × Bz k + Az k × Bx i + Az k × B y j + Az k × Bz k = Ax × Bx (i × i) + Ax × B y (i × j) + Ax × Bz (i × k) + Ay × Bx (j × i) + A y × B y (j × j) + Ay × Bz (j × k) + Az × Bx (k × i) + Az × B y (k × j) + Az × Bz (k × k) = Ax × Bx (0) + Ax × B y (k) + Ax × Bz (– j) + Ay × Bx (– k) + Ay × B y (0) + Ay × Bz (i) + Az × Bx (j) + Az × B y (– i) + Az × Bz (0) = i (Ay Bz – Az B y ) – j (Ax Bz – Az Bx ) + k (Ax B y – A y Bx ) x y z x y z i j k A A A B B B 