Let $\vec{A}=2 \hat{i}+\hat{k}, B=\hat{i}+\hat{j}+\hat{k}$ and $\vec{C}=4 \hat{i}-3 \hat{j}+7 \hat{k

Question

Let $\vec{A}=2 \hat{i}+\hat{k}, B=\hat{i}+\hat{j}+\hat{k}$ and $\vec{C}=4 \hat{i}-3 \hat{j}+7 \hat{k}$. Determine a vector $\vec{R}$ satisfying $\vec{R} 	imes \vec{B}=\vec{C} 	imes \vec{B}$ and $\vec{R} \cdot \vec{A}=0$\

Professor PC Thomas Classes · Answer

Let $\vec{R}=x \hat{i}+y \hat{j}+z \hat{k}$\ $\vec{R} 	imes \vec{B}=(y-z) \hat{i}+(z-x) \hat{j}+(x-y) \hat{k}$\ $\vec{C} 	imes \vec{B}=-10 \hat{i}+3 \hat{j}+7 \hat{k}$\ $\vec{R} \cdot \vec{A}=2 x+z$ Given $\vec{R} 	imes \vec{B}=\vec{C} 	imes \vec{B}$\ $	herefore \quad y-z=-10$\ $z-x=3$ and $x-y=7$\ $\vec{R} \cdot \vec{A}=0$\ $\Rightarrow \quad 2 x+z=0$\ On solving, we get $x=-1, y=-8, z=2$\ $	herefore \quad \vec{R}=-\hat{i}-8 \hat{j}+2 \hat{k}$ is the required vector.