Vector-Valued Functions

Core Sentence / 核心句

A vector-valued function packages coordinate functions into one time-dependent position.

$$\mathbf{r}(t)=⟨x(t),y(t)⟩,\mathbf{v}(t)={\mathbf{r}}^{\prime }(t),\mathbf{a}(t)={\mathbf{r}}^{\prime \prime }(t).$$ $$\text{speed}=\Vert \mathbf{v}(t)\Vert =\sqrt{[x^{\prime }(t){]}^2+[y^{\prime }(t){]}^2}.$$

Watch Out / 易错点

Velocity is a vector; speed is a scalar magnitude. Integrating velocity gives displacement, so final position still requires the initial position.

Connected Notes / 连接

• 微积分BC第12次正课-旋转体弧长参数向量与极坐标
• Parametric Equations
• Vectors
• One-Dimensional Motion
• Arc Length