# The vectors $\dpi{100} \vec{a}\; and \; \vec{b}$ are not perpendicular and $\dpi{100} \vec{c}\; and \; \vec{d}$ are two vectors satisfying : $\dpi{100} \vec{b}\; \times \; \vec{c}=\vec{b}\; \times \; \vec{d}$and $\dpi{100} \vec{a}\cdot \vec{d}=0$. Then the vector $\dpi{100} \vec{d}$ is equal to Option 1) $\vec{b}+\left ( \frac{\vec{b}\cdot \vec{c}}{\vec{a}\cdot \vec{b}} \right )\vec{c}$ Option 2) $\vec{c}-\left ( \frac{\vec{a}\cdot \vec{c}}{\vec{a}\cdot \vec{b}} \right )\vec{b}$ Option 3) $\vec{b}-\left ( \frac{\vec{b}\cdot \vec{c}}{\vec{a}\cdot \vec{b}} \right )\vec{c}$ Option 4) $\vec{c}+\left ( \frac{\vec{a}\cdot \vec{c}}{\vec{a}\cdot \vec{b}} \right )\vec{b}$

As we learnt in

Vector Triple Product (VTP) -

$\vec{a}\times \left ( \vec{b} \times \vec{c}\right )= \left ( \vec{a}.\vec{c} \right )\vec{b}-\left ( \vec{a}.\vec{b}\right )\vec{c}$

$\left ( \vec{a}\times \vec{b} \right )\times \vec{c}= \left ( \vec{a}.\vec{c} \right )\vec{b}-\left ( \vec{b}.\vec{c}\right )\vec{a}$

- wherein

$\vec{a}, \vec{b}, \vec{c}$are three vectors.

$\vec{b}\times \vec{d}\:=\:\vec{b}\times\vec{c}$

$(\vec{b}\times \vec{d})\times \vec{a}\:\:=(\vec{b}\times \vec{c})\times \vec{a}$

$(\vec{a}.\vec{b})\:\: \vec{d} - 0(\vec{b})=(\vec{a}.\vec{b})\vec{c}-(\vec{a}.\vec{c})\vec{b}$

$\\\vec{d}\:=\:\vec{c}-(\frac{\vec{a}.\vec{c}}{\vec{a}.\vec{b}})\vec{b}$

Option 1)

$\vec{b}+\left ( \frac{\vec{b}\cdot \vec{c}}{\vec{a}\cdot \vec{b}} \right )\vec{c}$

This option is incorrect.

Option 2)

$\vec{c}-\left ( \frac{\vec{a}\cdot \vec{c}}{\vec{a}\cdot \vec{b}} \right )\vec{b}$

This option is correct.

Option 3)

$\vec{b}-\left ( \frac{\vec{b}\cdot \vec{c}}{\vec{a}\cdot \vec{b}} \right )\vec{c}$

This option is incorrect.

Option 4)

$\vec{c}+\left ( \frac{\vec{a}\cdot \vec{c}}{\vec{a}\cdot \vec{b}} \right )\vec{b}$

This option is incorrect.

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