Question

The position vectors of four [point A, B, C and D are $\hat{i}+\hat{j}+\hat{k},2\hat{i}+5\hat{j},3\hat{i}+2\hat{j}-3\hat{k},\hat{i}-6\hat{j}-\hat{k}$ then select the correct relation:

• A. AB = CD
• B. $\text{AB}=\frac{1}{2}\text{CD}$
• C. $\text{AB}=\frac{2}{3}\text{CD}$
• D. $\text{AB}=\frac{1}{4}\text{CD}$

Answer 1

The correct option is B $\text{AB}=\frac{1}{2}\text{CD}$

Here, $AB=|\overrightarrow{B}-\overrightarrow{A}|=|(2\hat{i}+5\hat{j})-(\hat{i}+\hat{j}+\hat{k})|=|\hat{i}+4\hat{j}-\hat{k}|=\sqrt{1^2+4^2+(-1{)}^2}=\sqrt{18}=3\sqrt{2}$

Similarly, $CD=|\overrightarrow{B}-\overrightarrow{A}|=|(\hat{i}-6\hat{j}-\hat{k})-(3\hat{i}+2\hat{j}-3\hat{k})|=|-2\hat{i}-8\hat{j}+2\hat{k}|=\sqrt{(-2{)}^2+(-8{)}^2+(2{)}^2}=\sqrt{72}=6\sqrt{2}$

Thus, $AB=CD/2$