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