JEE MAIN - Mathematics (2002 - No. 32)
If the vectors $$\overrightarrow c ,\overrightarrow a = x\widehat i + y\widehat j + z\widehat k$$ and $$\widehat b = \widehat j$$ are such that $$\overrightarrow a ,\overrightarrow c $$ and $$\overrightarrow b $$ form a right handed system then $${\overrightarrow c }$$ is :
$$z\widehat i - x\widehat k$$
$$\overrightarrow 0 $$
$$y\widehat j$$
$$ - z\widehat i + x\widehat k$$
คำอธิบาย
Since $$\overrightarrow a ,\overrightarrow c ,\overrightarrow b $$ form a right handed system,
$$\therefore$$ $$\overrightarrow c = \overrightarrow b \times \overrightarrow a = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 0 & 1 & 0 \cr x & y & z \cr } } \right|$$
$$ = z\widehat i - x\widehat k$$
$$\therefore$$ $$\overrightarrow c = \overrightarrow b \times \overrightarrow a = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 0 & 1 & 0 \cr x & y & z \cr } } \right|$$
$$ = z\widehat i - x\widehat k$$