Question

Which of the following is an example of two different vectors with same magnitude?

• A. $(2\hat{i}+3\hat{j}+\hat{k})$ and $(2\hat{i}+3\hat{j}-\hat{k})$
• B. $(3\hat{i}+5\hat{j}+\hat{k})$ and $(3\hat{i}+4\hat{j}+\hat{k})$
• C. $(\hat{j}+\hat{k})$ and $(2\hat{j}+3\hat{k})$
• D. None of the above

Answer 1

Answer: (A)

Two vectors can have same magnitude, if the sum of the squares of coefficient of $\hat{i},\hat{j}$ and $\hat{k}$ is same. The vectors $a=(2\hat{i}+3\hat{j}+\hat{k})$ and $b=(2\hat{i}+3\hat{j}-\hat{k})$ are different vectors having the same magnitude.
Magnitude of Ist vector $=\sqrt{(2{)}^2+(3{)}^2+(1{)}^2}=\sqrt{4+9+1}=\sqrt{14}$
and magnitude of Ilnd vector$=\sqrt{(2{)}^2+(1{)}^2+(-3{)}^2}$
$=\sqrt{4+1+9}=\sqrt{14}$
i.e., they have same magnitude.