Question

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

Answer 1

Answer on Question #34254, Physics Electric Circuits

$\mathrm{A} \times \mathrm{B} = \mathrm{O}, \mathrm{B} \times \mathrm{C} = \mathrm{O}$ , and $\mathrm{A} \neq 0, \mathrm{B} \neq 0, \mathrm{C} \neq 0$ . Find the value of $\mathrm{A} \times \mathrm{C}$ .

Solution

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

Then

$$\begin{array}{l} \left[ \vec {B} \times \left[ \vec {C} \times \vec {A} \right] \right] = \vec {C} (\underbrace {\vec {B} \cdot \vec {A}} _ {\vec {B} | \vec {A}}) - \vec {A} \cdot (\underbrace {\vec {B} \cdot \vec {C}} _ {\vec {B} | \vec {C}}) = \vec {C} | \vec {B} | \cdot | \vec {A} | - \vec {A} \cdot | \vec {B} | \cdot | \vec {C} | = | \vec {B} | (\vec {C} | \vec {A} | - \vec {A} | \vec {C} |) = 0 \\ | \vec {B} | \neq 0 \\ \left(\vec {C} | \vec {A} | - \vec {A} | \vec {C} |\right) = 0 \Rightarrow \vec {C} \| \vec {A} \Rightarrow \left[ \vec {A} \times \vec {C} \right] = 0 \\ \end{array}$$

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