Find r(t) ·u(t) . Is the result a vector-valued function? Explain. 𝐫(t)=⟨3 cost, 2 sint, t-2⟩,

Name: Problem 18, Chapter 12: Calculus Early Transcendental Functions
Uploaded: 2021-07-23T14:02:17-08:00
Duration: 1 min 43 s
Channel: Monica Miller
Description: Find r(t) ·u(t) . Is the result a vector-valued function? Explain. 𝐫(t)=⟨3 cost, 2 sint, t-2⟩, 𝐮(t)=⟨4 sint,-6 cost, t^2⟩

step 1 We are going to be finding the dot product between R of T and U of T, and then we're also asked

Question

Find $r(t) \cdot u(t) .$ Is the result a vector-valued function? Explain. $$\mathbf{r}(t)=\langle 3 \cos t, 2 \sin t, t-2\rangle, \quad \mathbf{u}(t)=\left\langle 4 \sin t,-6 \cos t, t^{2}\right\rangle$$

   Find $r(t) \cdot u(t) .$ Is the result a vector-valued function? Explain.
$$\mathbf{r}(t)=\langle 3 \cos t, 2 \sin t, t-2\rangle, \quad \mathbf{u}(t)=\left\langle 4 \sin t,-6 \cos t, t^{2}\right\rangle$$

Calculus Early Transcendental Functions

Ron Larson, Bruce… 6th Edition

Chapter 12, Problem 18

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Solved by Expert Monica Miller

07/23/2021

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The dot product of two vectors is calculated by multiplying their corresponding components and then adding those products together. Show more…

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Find $r(t) \cdot u(t) .$ Is the result a vector-valued function? Explain. $$\mathbf{r}(t)=\langle 3 \cos t, 2 \sin t, t-2\rangle, \quad \mathbf{u}(t)=\left\langle 4 \sin t,-6 \cos t, t^{2}\right\rangle$$

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Key Concepts

Vector-Valued Function

A vector-valued function returns a vector for every input. Even though both operands in a dot product might be vector-valued functions, the dot product operation itself results in a scalar function, not another vector-valued function.

Dot Product

The dot product is an operation that takes two vectors and returns a scalar. It is computed by multiplying corresponding components of the two vectors and then summing the results.

Scalar Function

A scalar function assigns a single numerical value to each input, as opposed to a vector-valued function which assigns a vector. When a dot product is computed, the output is a scalar, regardless of the input vectors being functions.

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