Question

Let $\vec{ a }=2 \hat{i}-7 \hat{j}+5 \hat{k}, \vec{ b }=\hat{i}+\hat{k}$ and $\vec{ c }=\hat{i}+2 \hat{j}-3 \hat{k}$ be three given vectors. If $\vec{ r }$ is a vector such that $\vec{r} \times \vec{a}=\vec{c} \times \vec{a}$ and $\vec{r} \cdot \vec{b}=0$, then $|\vec{r}|$ is equal to :

• A. $\frac{11}{7} \sqrt{2}$
• B. $\frac{11}{7}$
• C. $\frac{\sqrt{914}}{7}$
• D. $\frac{11}{5} \sqrt{2}$

Answer 1

Answer: (A)

$ \vec{ a }=2 \hat{ i }-7 \hat{ j }+5 \hat{ k } $
$ \vec{ b }=\hat{ i }+\hat{ k } $
$ \vec{ c }=\hat{ i }+2 \hat{ j }-3 \hat{ k } $
$ \vec{ r } \times \vec{ a }=\vec{ c } \times \vec{ a } \Rightarrow(\vec{ r }-\vec{ c }) \times \vec{ a }=0 $
$ \therefore \vec{ r }=\vec{ c }+\lambda \vec{ a } $
$ \vec{ r } \cdot \vec{ b }=0 \Rightarrow \vec{ c } \cdot \vec{ b }+\lambda \quad \vec{ b } \cdot \vec{ a }=0 $
$ -2+\lambda(7)=0 \Rightarrow \lambda=\frac{2}{7} $
$ \therefore \vec{ r }=\vec{ c }+\frac{2 \vec{ a }}{7}=\frac{1}{7}(11 \hat{ i }-11 \hat{ k }) $
$ \mid \vec{ r }=\frac{11 \sqrt{2}}{7}$