If a =2 i +5 j -7 k , b =-3 i +4 j + k and c = i -2 j -3 k , compute ( a b ) c and a ( b c ) and verify that these are not equal.

Given: a =2 i +5 j -7 k b =-3 i +4 j + k c = i -2 j -3 k a b = i & j & k 2&5&-7 -3&4&1 =(5+28) i -(2-21) j +(8+15) k =33 i +19 j +23 k ( a b ) c = i & j & k 33&19&23 1&-2&-3 =(-57+46) i -(-99-23) j +(-66-19) k ( a b ) c =-11 i +122 j -85 k (1) b c = i & j & k -3&4&1 1&-2&-3 =(-12+2) i -(9-1) j +(6-4) k =-10 i -8 j +2 k a ( b c )= i & j & k 2&5&-7 -10&-8&2 =(10-56) i -(4-70) j +(-16+50) k a ( b c )=-46 i +66 j +34 k (2) From (1) and (2), we get ( a b ) c a ( b c )

If a =2 i +5 j -7 k , b =-3 i +4 j + k and c = i -2 j -3 k , comp…

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Question

If $\vec{\text{a}}=2\hat{\text{i}}+5\hat{\text{j}}-7\hat{\text{k}},\vec{\text{b}}=-3\hat{\text{i}}+4\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}},$ compute $\big(\vec{\text{a}}\times\vec{\text{b}}\big)\times\vec{\text{c}}$ and $\vec{\text{a}}\times\big(\vec{\text{b}}\times\vec{\text{c}}\big)$ and verify that these are not equal.

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Answer

Given:
$\vec{\text{a}}=2\hat{\text{i}}+5\hat{\text{j}}-7\hat{\text{k}}$
$\vec{\text{b}}=-3\hat{\text{i}}+4\hat{\text{j}}+\hat{\text{k}}$
$\vec{\text{c}}=\hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}}$
$\therefore\vec{\text{a}}\times\vec{\text{b}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\2&5&-7\\-3&4&1 \end{vmatrix}$
$=(5+28)\hat{\text{i}}-(2-21)\hat{\text{j}}+(8+15)\hat{\text{k}}$
$=33\hat{\text{i}}+19\hat{\text{j}}+23\hat{\text{k}}$
$\Rightarrow\big(\vec{\text{a}}\times\vec{\text{b}}\big)\times\vec{\text{c}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\33&19&23\\1&-2&-3 \end{vmatrix}$
$=(-57+46)\hat{\text{i}}-(-99-23)\hat{\text{j}}+(-66-19)\hat{\text{k}}$
$\Rightarrow\big(\vec{\text{a}}\times\vec{\text{b}}\big)\times\vec{\text{c}}=-11\hat{\text{i}}+122\hat{\text{j}}-85\hat{\text{k}}\dots(1)$
$\therefore\vec{\text{b}}\times\vec{\text{c}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\-3&4&1\\1&-2&-3 \end{vmatrix}$
$=(-12+2)\hat{\text{i}}-(9-1)\hat{\text{j}}+(6-4)\hat{\text{k}}$
$=-10\hat{\text{i}}-8\hat{\text{j}}+2\hat{\text{k}}$
$\Rightarrow\vec{\text{a}}\times\big(\vec{\text{b}}\times\vec{\text{c}}\big)=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\2&5&-7\\-10&-8&2 \end{vmatrix}$
$=(10-56)\hat{\text{i}}-(4-70)\hat{\text{j}}+(-16+50)\hat{\text{k}}$
$\Rightarrow\vec{\text{a}}\times\big(\vec{\text{b}}\times\vec{\text{c}}\big)=-46\hat{\text{i}}+66\hat{\text{j}}+34\hat{\text{k}}\dots(2)$
From (1) and (2), we get
$\big(\vec{\text{a}}\times\vec{\text{b}}\big)\times\vec{\text{c}}\neq\vec{\text{a}}\times\big(\vec{\text{b}}\times\vec{\text{c}}\big)$