Question

Prove that ax(bxc)+bx(cxa)+cx(axb)=0 in vector formula

Answer 1

$a\times (b\times c)+b\times(c\times a)+c\times(a\times b)=0$, proved.

Step-by-step explanation:

Prove that, $a\times (b\times c)+b\times(c\times a)+c\times(a\times b)=0$

L.H.S.$=a\times (b\times c)+b\times(c\times a)+c\times(a\times b)$

$=a\times (a)+b\times(b)+c\times(c)$

[∵ $b\times c=a,c\times a=b$ and $a\times b=c$]

$=a\times a+b\times b+c\times c$

= 0 + 0 + 0

[∵ $a\times a=0,b\times b=0$ and $c\times c=0$]

= 0

= R.H.S., proved.

Hence, $a\times (b\times c)+b\times(c\times a)+c\times(a\times b)=0$, proved.

Answer 2

We have to prove prove that, $a\times (b\times c)+b\times(c\times a)+c\times(a\times b)=0$

Solution:

L.H.S.= $a\times (b\times c)+b\times(c\times a)+c\times(a\times b)$

$=a\times (a)+b\times(b)+c\times(c)$

Using the vector identity:

$b\times c=a,c\times a=b$ and $a\times b=c$

= $a\times a$ + $b\times b$ + $c\times c$

= 0 + 0 + 0

Using the vector identity:

$a\times a$ = 0, $b\times b$ = 0 and $c\times c$ = 0

= 0

= R.H.S., proved.

Thus, $a\times (b\times c)+b\times(c\times a)+c\times(a\times b)=0$, proved.