Question
Give an example for which $\overrightarrow{\text{A}}.\overrightarrow{\text{B}}=\overrightarrow{\text{C}}.\overrightarrow{\text{B}}$ but $\overrightarrow{\text{A}}\neq\overrightarrow{\text{C}}.$
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Answer
For example, as shown in the figure,$\overrightarrow{\text{A}}\perp\overrightarrow{\text{B}}:\overrightarrow{\text{B}} \ \text{along west}$
$\overrightarrow{\text{B}}\perp\overrightarrow{\text{C}}:\overrightarrow{\text{A}}\text{ along south}$
$\overrightarrow{\text{C}} \text{along north}$
$\overrightarrow{\text{A}}.\overrightarrow{\text{B}}=0$ $\therefore\overrightarrow{\text{A}}.\overrightarrow{\text{B}}=\overrightarrow{\text{B}}.\overrightarrow{\text{C}}$
$\overrightarrow{\text{B}}.\overrightarrow{\text{C}}=0$ But $\overrightarrow{\text{B}}\neq\overrightarrow{\text{C}}$
$\overrightarrow{\text{B}}\perp\overrightarrow{\text{C}}:\overrightarrow{\text{A}}\text{ along south}$
$\overrightarrow{\text{C}} \text{along north}$
$\overrightarrow{\text{A}}.\overrightarrow{\text{B}}=0$ $\therefore\overrightarrow{\text{A}}.\overrightarrow{\text{B}}=\overrightarrow{\text{B}}.\overrightarrow{\text{C}}$
$\overrightarrow{\text{B}}.\overrightarrow{\text{C}}=0$ But $\overrightarrow{\text{B}}\neq\overrightarrow{\text{C}}$
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