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Chapter 11: Problem 1

Interpret the following statement: Points have a location, but no size or direction; nonzero vectors have a size and direction, but no location.

Short Answer

Expert verified
Answer: The main differences between points and nonzero vectors are that points have a location (coordinates) but no size or direction, while nonzero vectors have size (magnitude) and direction, but no fixed location. They can be shifted anywhere in space and still maintain their magnitude and direction.

Step by step solution

01

Understanding Points

A point is a basic element of geometry that represents a location in space. Points are represented by coordinates (e.g., x and y coordinates in two-dimensional space). These coordinates give the position of a point, but it does not have any size or direction.
02

Understanding Nonzero Vectors

A nonzero vector is a mathematical entity that represents direction and magnitude. It can be represented as an arrow with a starting point and an endpoint. Nonzero vectors have size (also called magnitude) and direction, but they do not have a fixed location.
03

Comparing Points and Nonzero Vectors

To summarize the differences between points and nonzero vectors: 1. Points have a location (coordinates), but no size or direction. 2. Nonzero vectors have size (magnitude) and direction, but no fixed location. They can be shifted anywhere in space and still maintain their magnitude and direction.

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Most popular questions from this chapter

Derivative rules Suppose \(\mathbf{u}\) and \(\mathbf{v}\) are differentiable functions at \(t=0\) with \(\mathbf{u}(0)=\langle 0,1,1\rangle, \mathbf{u}^{\prime}(0)=\langle 0,7,1\rangle\) \(\mathbf{v}(0)=\langle 0,1,1\rangle,\) and \(\mathbf{v}^{\prime}(0)=\langle 1,1,2\rangle .\) Evaluate the following expressions. a. \(\left.\frac{d}{d t}(\mathbf{u} \cdot \mathbf{v})\right|_{t=0}\) b. \(\left.\frac{d}{d t}(\mathbf{u} \times \mathbf{v})\right|_{t=0}\) c. \(\left.\frac{d}{d t}(\cos t \mathbf{u}(t))\right|_{t=0}\)

An object moves along a path given by $$\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle$$ for \(0 \leq t \leq 2 \pi\) a. Show that the curve described by \(\mathbf{r}\) lies in a plane. b. What conditions on \(a, b, c, d, e,\) and \(f\) guarantee that the curve described by \(\mathbf{r}\) is a circle?

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. Algebra inequality Show that $$\left(u_{1}+u_{2}+u_{3}\right)^{2} \leq 3\left(u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\right)$$ for any real numbers \(u_{1}, u_{2},\) and \(u_{3} .\) (Hint: Use the CauchySchwarz Inequality in three dimensions with \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) and choose v in the right way.)

Diagonals of a parallelogram Consider the parallelogram with adjacent sides \(\mathbf{u}\) and \(\mathbf{v}\) a. Show that the diagonals of the parallelogram are \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}-\mathbf{v}\) b. Prove that the diagonals have the same length if and only if \(\mathbf{u} \cdot \mathbf{v}=0\) c. Show that the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the sides.

Direction angles and cosines Let \(\mathbf{v}=\langle a, b, c\rangle\) and let \(\alpha, \beta\) and \(\gamma\) be the angles between \(\mathbf{v}\) and the positive \(x\) -axis, the positive \(y\) -axis, and the positive \(z\) -axis, respectively (see figure). a. Prove that \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1\) b. Find a vector that makes a \(45^{\circ}\) angle with \(\mathbf{i}\) and \(\mathbf{j}\). What angle does it make with \(\mathbf{k} ?\) c. Find a vector that makes a \(60^{\circ}\) angle with i and \(\mathbf{j}\). What angle does it make with k? d. Is there a vector that makes a \(30^{\circ}\) angle with \(\mathbf{i}\) and \(\mathbf{j} ?\) Explain. e. Find a vector \(\mathbf{v}\) such that \(\alpha=\beta=\gamma .\) What is the angle?
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