Question

The vectors $\vec{a}$ and $\vec{b}$ are not perpendicular and $\vec{c}$ and $\vec{d}$ are two vectors satisfying $\vec{b} \times \vec{c}=\vec{b} \times \vec{d}$ and $\vec{a} \cdot \vec{d}=0 .$ Then the vector $\vec{d}$ is equal to

• A. $\bar{c}-\left(\frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}}\right) \vec{b}$
• B. $\vec{b}-\left(\frac{\vec{b} \cdot \vec{c}}{\vec{a} \cdot \vec{b}}\right) \vec{c}$
• C. $\vec{c}+\left(\frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}}\right) \vec{b}$
• D. $\vec{b}+\left(\frac{\vec{b} \cdot \vec{c}}{\vec{a} \cdot \vec{b}}\right) \vec{c}$

Answer 1

Answer: (A)

We have
$ \vec{b} \times \vec{c}=\vec{b} \times \vec{d}$
$\Rightarrow \vec{a} \times(\vec{b} \times \bar{c})=\vec{a} \times(\vec{b} \times \bar{d}) $
$\Rightarrow (\vec{a} \cdot \vec{c}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{c}=(\vec{a} \cdot \vec{d}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{d}$
$ \Rightarrow (\vec{a} \cdot \vec{b}) \vec{d}=-(\vec{a} \cdot \vec{c}) \vec{b}+(\vec{a} \cdot \vec{b}) \vec{c} $
$ \Rightarrow \vec{d}=-\frac{(\vec{a} \cdot \vec{c})}{(\vec{a} \cdot \vec{b})} \vec{b}+\vec{c} $