Let $\phi_1, \phi_2: \mathbb{R} \rightarrow \mathbb{R}$ be non-constant and differentiable functions. Define $\vec{v}=$ $\phi_1(t) \hat{i}+\phi_2(t) \hat{j}$ and $\frac{d \vec{v}}{d t}=\frac{d \phi_1(t)}{d t} \hat{i}+\frac{d \phi_2(t)}{d t} \hat{j}$. Then

• A

  $\frac{d \vec{v}}{d t}=\overrightarrow{0}$
• Correct answer
  B

  $\vec{v} \cdot \frac{d \vec{v}}{d t}=0 \Longrightarrow|\vec{v}|$ is constant
• Correct answer
  C

  $|\vec{v}|$ is constant $\Longrightarrow \vec{v} \cdot \frac{d \vec{v}}{d t}=0$
• D

  $\left|\frac{d \vec{v}}{d t}\right|$ is not constant for any choice of $\phi_1$ and $\phi_2$

Hence, the correct answer is options: 'B', 'C'.