Let $\vec{a} = \hat{i} + 2\hat{j}$ and $\vec{b} = 2\hat{i} + \hat{j}$. Is $|\vec{a}| = |\vec{b}|$? Are the vectors $\vec{a}$ and $\vec{b}$ equal?

(N/A) Given vectors are $\vec{a} = \hat{i} + 2\hat{j}$ and $\vec{b} = 2\hat{i} + \hat{j}$.
The magnitude of vector $\vec{a}$ is $|\vec{a}| = \sqrt{1^{2} + 2^{2}} = \sqrt{1 + 4} = \sqrt{5}$.
The magnitude of vector $\vec{b}$ is $|\vec{b}| = \sqrt{2^{2} + 1^{2}} = \sqrt{4 + 1} = \sqrt{5}$.
Since $|\vec{a}| = \sqrt{5}$ and $|\vec{b}| = \sqrt{5}$,we have $|\vec{a}| = |\vec{b}|$.
Two vectors are equal if and only if their corresponding components are equal. Here,the components of $\vec{a}$ are $(1, 2)$ and the components of $\vec{b}$ are $(2, 1)$.
Since $(1, 2) \neq (2, 1)$,the vectors $\vec{a}$ and $\vec{b}$ are not equal.