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3-1.Vectors
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If $\overrightarrow{ F }=2 \hat{ i }+\hat{ j }-\hat{ k }$ and $\overrightarrow{ r }=3 \hat{ i }+2 \hat{ j }-2 \hat{ k }$, then the scalar and vector products of $\overrightarrow{ F }$ and $\overrightarrow{ r }$ have the magnitudes respectively as
A
$5, \sqrt{3}$
B
$4, \sqrt{5}$
C
$10, \sqrt{2}$
D
$10,2$
(NEET-2022)
Solution
$\overrightarrow{ F }=2 \hat{ i }+\hat{ j }-\hat{ k }$ $\quad \overrightarrow{ r }=3 \hat{ i }+2 \hat{ j }-2 \hat{ k }$
$\overrightarrow{ F } \cdot \overrightarrow{ r }=6+2+2=10$
$\overrightarrow{ F } \times \overrightarrow{ r }=\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 1 & -1 \\ 3 & 2 & -2\end{array}\right|$
$=\hat{ i }(0)-\hat{ j }(-1)+\hat{ k }(1)$
$=\hat{ j }+\hat{ k }$
$|\overrightarrow{ F } \times \overrightarrow{ r }|=\sqrt{2}$
Standard 11
Physics
