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.