82.3 the Cartesian representation of any vector where ixi=0 ixj=k ixk=-7 x7=-×1=0xk k×l= kxj=-1k×k=0 5. Variation of a vector O The Magnitude changes, the direction is preserved; 3 The direction changes, the magnitude is preserved; 3 Both the magnitude and direction change. 4A A Discuss①and② A1+ 8 2.3 the Cartesian representation of any vector Differentiation of a vector. da d dA d dA dt dt (AI+A, j+ A k) 十 十 da dB (4+B) 十 da B+A dB d da dB A×B) xB+Ax d 88 §2.3 the Cartesian representation of any vector where 0 ˆ ˆ i × i = j i k ˆ ˆ ˆ × = − k i j ˆ ˆ ˆ × = i j k ˆ ˆ ˆ × = 0 ˆ ˆ j × j = k j i ˆ ˆ ˆ × = − i k j ˆ ˆ ˆ × = − j k i ˆ ˆ × = r 0 ˆ ˆ k × k = 5. Variation of a vector 1 The Magnitude changes, the direction is preserved; 2 The direction changes, the magnitude is preserved; 3 Both the magnitude and direction change. Ai r Af r A r ∆ A A A A A A f i f i r r r r r r ∆ ∆ = + = − Discuss 1 and 2 §2.3 the Cartesian representation of any vector t B t A A B t d d d d ( ) d d r r r r + = + t B B A t A A B t d d d d ( ) d d r r r r r r ⋅ = ⋅ + ⋅ t B B A t A A B t d d d d ( ) d d r r r r r r × = × + × Differentiation of a vector: k t A j t A i t A A i A j A k t t A x y z x y z ˆ d d ˆ d d ˆ d d ) ˆ ˆ ˆ ( d d d d = + + = + + r