Let 𝐫1(t)=⟨t^2, 1,2 t⟩, 𝐫2(t)=⟨1,2, e^t⟩Compute (d)/(d t) 𝐫1(t) ·𝐫2.(t)|t=1 in two ways: (a) C

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Let 𝐫1(t)=⟨t^2, 1,2 t⟩, 𝐫2(t)=⟨1,2, e^t⟩Compute (d)/(d t)...

Let $$ \mathbf{r}_{1}(t)=\left\langle t^{2}, 1,2 t\right\rangle, \quad \mathbf{r}_{2}(t)=\left\langle 1,2, e^{t}\right\rangle $$ Compute $\frac{d}{d t} \mathbf{r}_{1}(t) \cdot \mathbf{r}_{2}\left.(t)\right|_{t=1}$ in two ways: (a) Calculate $\mathbf{r}_{1}(t) \cdot \mathbf{r}_{2}(t)$ and differentiate. (b) Use the Product Rule.

Let $$
\mathbf{r}_{1}(t)=\left\langle t^{2}, 1,2 t\right\rangle, \quad \mathbf{r}_{2}(t)=\left\langle 1,2, e^{t}\right\rangle
$$
Compute $\frac{d}{d t} \mathbf{r}_{1}(t) \cdot \mathbf{r}_{2}\left.(t)\right|_{t=1}$ in two ways: (a) Calculate $\mathbf{r}_{1}(t) \cdot \mathbf{r}_{2}(t)$ and differentiate. (b) Use the Product Rule.

Key Concepts

Vector-Valued Functions

A vector-valued function assigns a vector to each value of a parameter, typically representing curves or paths in space. In this context, the functions r1(t) and r2(t) express motions or positions in three dimensions, and understanding their component forms is crucial for performing operations like differentiation and the dot product.

Dot Product

The dot product is an operation that takes two vectors and returns a scalar. It is computed by multiplying corresponding components and summing the products. The dot product is fundamental in determining angles between vectors and in projecting one vector onto another, and its properties are used in the differentiation process for scalar functions derived from vector dot products.

Differentiation of Vector Functions

Differentiating vector functions involves taking the derivative of each component separately. This process is essential for analyzing the rates of change of vector-valued functions, such as determining velocity from a position vector. Component-wise differentiation is the basis for applying further rules like the product rule when working with dot products.

Product Rule for Dot Products

The product rule for the dot product states that the derivative of the dot product of two vector functions is the dot product of the derivative of the first function with the second plus the dot product of the first function with the derivative of the second. This rule allows one to differentiate expressions involving dot products efficiently by taking advantage of the linearity of differentiation.

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