Question

prove 13-16 which are often useful in practical work and ilustrate each formula with two example Hint: for 13 choose cartezian coordinate such that d=[d1,0,0,]and c=[c1,c2,0].show that each side of(13) then equals [-b2c2d1,b1c2d1,0]and give reasons why the two sides are then equal in any Cartesian coordinatesystem.for 14 and 15 use 13.

13.bx(cxd)=(b.d)c-(b.c)b

14.(axb)x(cxd)=(a b d)c-(a b c)d

15.(axb).(cxd)=(a.c)(b.d)-(a..d)(b.c)

16.(a b c)=(b c a)=(c a b)=-(c b a)=-(c b a)=-(a c b)

Answer 1

To prove Rightarrow (a times b) times (c times d) = (abd)c - (abc)d let (a times b) = 1, then 1 times (c times d) = (1 middot d)c - (1 middot c)d = [(a times b) middot d]c - [(a times b) middot c]d (a times b) times (c times d) = [abd]c - [abc]d To prove Rightarrow (a times b) middot (c times d) = (a middot c)(b middot d) - (a middot d)(b middot c) let a times b = 1 1 middot (c times d) = (1 times c) middot d = [(a times b) times c] middot d = [(a middot c)b - a(b middot c)]d (a middot c) (b middot d) - (a middot d) (b middot c) To prove Rightarrow b times (c times d) = (b middot d)c - (b middot c)d {b times (c times d)} = b_1(C_xd_y - C_yd_x) - b_2(C_zd_x - C_xd_z)