Question

(b) Prove that ſă +5,6 +ē,č + ā] = 2ļābē]

11th - 12th Class

Maths

Solution

4.0 (1 ratings)

i) By definition ( [a, b, c]=a(b times c) ) ii) Also by property, ( mid a, b, c]=[b, c, a]=[c, a, b], ) taking in the same clock wise order, however if the order is reversed, then they are not equal; that is ( [a, b, c] eq[a, c, b] ) 2) Applying the above, ( mid a+b, b+c, c+a]=(a+b){(b+c) times(c+a)} ) ( =(a+b) cdot((b times c)+(b times a)+(c times c)+(c times a)) ) [Cross product of vectors is distributive over addition] ( =(a+b) cdot((b times c)+(b times a)+(c times a))[operatorname{sincec} x c=0] ) ( =a(b times c)+a(b times a)+a(c times a)+b cdot(b times c)+b cdot(b times a)+b cdot(c times a) ) ( =[a, b, c]+[a, b, a]+[a, c, a]+[b, b, c]+[b, c, a] ) ( =[a, b, c]+[a, b, c] text { isince }[a, b, a][a, c, a] text { and }[b, b c] e a c h=0 ; text { they form coplanar vectors }) ) ( =[a, b, c]+|a, b, c| text { From } 1, text { (ii) above }} ) ( =2 mid a, b, c] ) Thus it is proved that ( [a+b, b+c, c+a]=2[a b c] )