Vector Algebra Question 98
Question: Let $ {{\vec{r}}_1},{{\vec{r}}2},{{\vec{r}}3},…..{{\vec{r}}{n}}, $ be the position vectors of points $ P_1,P_2,P_3,….,P{n} $ relative to the origin O. If the vector equation $ a_1{{\vec{r}}1}+a_2{{\vec{r}}2}+….+a{n}{{\vec{r}}{n}}=0 $ holds, then a similar equation will also hold w.r.t. to any other origin provided
Options:
A) $ a_1+a_2+….+a_{n}=n $
B) $ a_1+a_2+….+a_{n}=1 $
C) $ a_1+a_2+….+a_{n}=0 $
D) $ a_1=a_2=a_3=….=a_{n}=0 $
Show Answer
Answer:
Correct Answer: C
Solution:
- [c] Given $ a_1{{\vec{r}}1}+a_2{{\vec{r}}2}+…..+a{n}{{\vec{r}}{n}}=0 $ Now $ \vec{a}+{{\vec{r}}1}’={{\vec{r}}1} $ and so on Hence, $ a_1(\vec{a}+{{\vec{r}}1})+a_2(\vec{a}+{{\vec{r}}2})+….+a{n}(\vec{a}+{{\vec{r}}{n}}’)=0 $ $ a_1{{\vec{r}}1}’+a_2{{\vec{r}}2}’+….+a{n}{{\vec{r}}{n}}’+\vec{a}(a_1+a_2+….+a{n})=0 $ Hence, $ a_1{{\vec{r}}1}’+a_2{{\vec{r}}2}’+….+a{n}{{\vec{r}}{n}}’=0 $ if $ a_1+a_2 $ $ +….+a{n}=0 $ .
BETA
