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  4. ABCDEF is a regular hexagon where centre O is the origin. If the position vectors of A and B are hat i - hat j +2 hat k and 2 hat i + hat j - hat k respectively, then BC is equal to

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Q. ABCDEF is a regular hexagon where centre $O$ is the origin. If the position vectors of $A$ and $B$ are $\hat{ i }-\hat{ j }+2 \hat{ k }$ and $2 \hat{ i }+\hat{ j }-\hat{ k }$ respectively, then $\overrightarrow{ BC }$ is equal to

Vector Algebra

Solution:

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$\overrightarrow{ OA }=\hat{ i }-\hat{ j }+2 \hat{ k }, \overrightarrow{ OB }=2 \hat{ i }+\hat{ j }-\hat{ k }$
$\overrightarrow{ OC }=\overrightarrow{ AB }=\overrightarrow{ OB }-\overrightarrow{ OA }=\hat{ i }+2 \hat{ j }-3 \hat{ k }$
$\overrightarrow{ BC }=\overrightarrow{ OC }-\overrightarrow{ OB }=(\hat{ i }+2 \hat{ j }-3 \hat{ k })-(2 \hat{ i }+\hat{ j }-\hat{ k })$
$=-\hat{ i }+\hat{ j }-2 \hat{ k }$