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Chapter 11: Problem 2

Express the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) in terms of the components of the vectors.

Short Answer

Expert verified
Answer: The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) can be found using the formula: \(\mathbf{u} \cdot \mathbf{v} = u_{1}v_{1} + u_{2}v_{2} + \cdots + u_{n}v_{n}\) where \(u_i\) and \(v_i\) represent the components of the vectors \(\mathbf{u}\) and \(\mathbf{v}\), respectively.

Step by step solution

01

Understand what a dot product is

The dot product (also known as the scalar product or inner product) is an operation that takes two vectors and returns a scalar value. This value is a measure of the similarity between the two vectors. If the dot product of two vectors is equal to 0, it means that the vectors are orthogonal (perpendicular) to each other.
02

Introduce the vectors

Let \(\mathbf{u}\) and \(\mathbf{v}\) be two arbitrary vectors in \(\mathbb{R}^n\) such that: \(\mathbf{u} = \begin{pmatrix} u_{1}\\ u_{2}\\ \vdots\\ u_{n} \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} v_{1}\\ v_{2}\\ \vdots\\ v_{n} \end{pmatrix}\)
03

Calculate the dot product

The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is defined as: \(\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^{n} u_{i}v_{i}\) where \(u_i\) and \(v_i\) represent the components of the vectors \(\mathbf{u}\) and \(\mathbf{v}\), respectively.
04

Write the final expression

Now, we can express the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) in terms of the components of the vectors: \(\mathbf{u} \cdot \mathbf{v} = u_{1}v_{1} + u_{2}v_{2} + \cdots + u_{n}v_{n}\)

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Most popular questions from this chapter

Cusps and noncusps a. Graph the curve \(\mathbf{r}(t)=\left\langle t^{3}, t^{3}\right\rangle .\) Show that \(\mathbf{r}^{\prime}(0)=\mathbf{0}\) and the curve does not have a cusp at \(t=0 .\) Explain. b. Graph the curve \(\mathbf{r}(t)=\left\langle t^{3}, t^{2}\right\rangle .\) Show that \(\mathbf{r}^{\prime}(0)=\mathbf{0}\) and the curve has a cusp at \(t=0 .\) Explain. c. The functions \(\mathbf{r}(t)=\left\langle t, t^{2}\right\rangle\) and \(\mathbf{p}(t)=\left\langle t^{2}, t^{4}\right\rangle\) both satisfy \(y=x^{2} .\) Explain how the curves they parameterize are different. d. Consider the curve \(\mathbf{r}(t)=\left\langle t^{m}, t^{n}\right\rangle,\) where \(m>1\) and \(n>1\) are integers with no common factors. Is it true that the curve has a cusp at \(t=0\) if one (not both) of \(m\) and \(n\) is even? Explain.

Proof of Product Rule By expressing \(\mathbf{u}\) in terms of its components, prove that $$\frac{d}{d t}(f(t) \mathbf{u}(t))=f^{\prime}(t) \mathbf{u}(t)+f(t) \mathbf{u}^{\prime}(t)$$

Compute the following derivatives. $$\frac{d}{d t}\left(\left(3 t^{2} \mathbf{i}+\sqrt{t} \mathbf{j}-2 t^{-1} \mathbf{k}\right) \cdot(\cos t \mathbf{i}+\sin 2 t \mathbf{j}-3 t \mathbf{k})\right)$$

Find the function \(\mathbf{r}\) that satisfies the given conditions. $$\mathbf{r}^{\prime}(t)=\langle\sqrt{t}, \cos \pi t, 4 / t\rangle ; \mathbf{r}(1)=\langle 2,3,4\rangle$$

Cauchy-Schwarz Inequality The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\) (because \(|\cos \theta| \leq 1\) ). This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. Triangle Inequality Consider the vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{u}+\mathbf{v}\) (in any number of dimensions). Use the following steps to prove that \(|\mathbf{u}+\mathbf{v}| \leq|\mathbf{u}|+|\mathbf{v}|\) a. Show that \(|\mathbf{u}+\mathbf{v}|^{2}=(\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v})=|\mathbf{u}|^{2}+\) \(2 \mathbf{u} \cdot \mathbf{v}+|\mathbf{v}|^{2}\) b. Use the Cauchy-Schwarz Inequality to show that \(|\mathbf{u}+\mathbf{v}|^{2} \leq(|\mathbf{u}|+|\mathbf{v}|)^{2}\) c. Conclude that \(|\mathbf{u}+\mathbf{v}| \leq|\mathbf{u}|+|\mathbf{v}|\) d. Interpret the Triangle Inequality geometrically in \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3}\).
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