Question #34254

A × B =O,B × C=O, and A≠0‚B≠0,C≠0.Find the value of A × C.

Expert's answer

Answer on Question #34254, Physics Electric Circuits

A×B=O,B×C=O\mathrm{A} \times \mathrm{B} = \mathrm{O}, \mathrm{B} \times \mathrm{C} = \mathrm{O} , and A0,B0,C0\mathrm{A} \neq 0, \mathrm{B} \neq 0, \mathrm{C} \neq 0 . Find the value of A×C\mathrm{A} \times \mathrm{C} .

Solution

[A×[B×C]=0]+[B×[C×A]]+[C×[A×B]0]=0[A×0]+[B×[C×A]]+[C×0]=0[B×[C×A]]=0[B×[A×C]]=0\begin{array}{l} \left[ \vec {A} \times \underbrace {\left[ \vec {B} \times \vec {C} \right]} _ {= 0} \right] + \left[ \vec {B} \times \left[ \vec {C} \times \vec {A} \right] \right] + \left[ \vec {C} \times \underbrace {\left[ \vec {A} \times \vec {B} \right]} _ {0} \right] = 0 \Rightarrow \left[ \vec {A} \times \vec {0} \right] + \left[ \vec {B} \times \left[ \vec {C} \times \vec {A} \right] \right] + \left[ \vec {C} \times \vec {0} \right] = 0 \\ \left[ \vec {B} \times \left[ \vec {C} \times \vec {A} \right] \right] = 0 \Rightarrow - \left[ \vec {B} \times \left[ \vec {A} \times \vec {C} \right] \right] = 0 \\ \end{array}


Then


[B×[C×A]]=C(BABA)A(BCBC)=CBAABC=B(CAAC)=0B0(CAAC)=0CA[A×C]=0\begin{array}{l} \left[ \vec {B} \times \left[ \vec {C} \times \vec {A} \right] \right] = \vec {C} (\underbrace {\vec {B} \cdot \vec {A}} _ {\vec {B} | \vec {A}}) - \vec {A} \cdot (\underbrace {\vec {B} \cdot \vec {C}} _ {\vec {B} | \vec {C}}) = \vec {C} | \vec {B} | \cdot | \vec {A} | - \vec {A} \cdot | \vec {B} | \cdot | \vec {C} | = | \vec {B} | (\vec {C} | \vec {A} | - \vec {A} | \vec {C} |) = 0 \\ | \vec {B} | \neq 0 \\ \left(\vec {C} | \vec {A} | - \vec {A} | \vec {C} |\right) = 0 \Rightarrow \vec {C} \| \vec {A} \Rightarrow \left[ \vec {A} \times \vec {C} \right] = 0 \\ \end{array}


Answer: [A×C]=0\left[\vec{A} \times \vec{C}\right] = 0

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