Value of hat{ i } ×(hat{ i } × a )+hat{ j } ×(hat{ j } × a )+hat{ k } ×(hat{ k } × a ) is

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3-1.Vectors

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The value of $\hat{ i } \times(\hat{ i } \times a )+\hat{ j } \times(\hat{ j } \times a )+\hat{ k } \times(\hat{ k } \times a )$ is

A$a$

B$a \times \hat{ k }$

C$-2\,a$

D$-a$

Solution

(c)
Suppose $a =a_1 \hat{ i }+a_2 \hat{ j }+a_3 \hat{ k }$
Now, $\quad(\hat{ i } \times a )=a_2 \hat{ k }-a_3 \hat{ j }$
Now, $\quad \hat{ i } \times(\hat{ i } \times a )=-a_2 \hat{ j }-a_3 \hat{ k }$
Similarly, $\quad \hat{ j } \times(\hat{ j } \times a )=-a_1 \hat{ i }-a_3 \hat{ k }$
and $\quad \hat{ k } \times(\hat{ j } \times a )=-a_1 \hat{ i }-a_2 \hat{ j }$
$\therefore \quad \hat{ i }(\hat{ i } \times a )+\hat{ j }(\hat{ j } \times a )+\hat{ k }(\hat{ k } \times a )=-2 a$

Standard 11

Physics

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The value of $\hat{ i } \times(\hat{ i } \times a )+\hat{ j } \times(\hat{ j } \times a )+\hat{ k } \times(\hat{ k } \times a )$ is

Solution

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