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Chapter 12: 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

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Answer: Points have a location, but no size or direction, and represent fixed positions in space defined by coordinates. Nonzero vectors have a size and direction, but no location, representing magnitudes and orientations that can be moved freely in space without changing their characteristics.

Step by step solution

01

Understanding Points

Points are fundamental elements in geometry, representing a specific location in space. Think of them as tiny dots in a plane or space, with a specific position defined by coordinates. However, points do not possess any size (i.e., length, width, or height) or direction. Remember, points are used to represent locations, not dimensions or movement.
02

Understanding Nonzero Vectors

A vector is a mathematical object that possesses both a magnitude (size) and a direction. Nonzero vectors are vectors whose magnitude is not zero, meaning they have a length and a specific orientation. They are usually represented by arrows, with the arrow length indicating the size, and the arrow direction indicating the orientation. Unlike points, vectors do not have a fixed location. They can be moved freely without altering their size or direction.
03

Comparing Points and Nonzero Vectors

To summarize the differences between points and nonzero vectors: - Points have a location, but no size or direction. They represent fixed positions in space defined by coordinates. - Nonzero vectors have a size and direction, but no location. They represent magnitudes and orientations, and can be moved freely in space without changing their characteristics. By understanding the properties of points and nonzero vectors, we can better grasp their functions and applications in various mathematical contexts.

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Most popular questions from this chapter

Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle\), and \(\mathbf{w}=\) \(\left\langle w_{1}, w_{2}, w_{3}\right\rangle\). Let \(c\) be a scalar. Prove the following vector properties. \(c(\mathbf{u} \cdot \mathbf{v})=(c \mathbf{u}) \cdot \mathbf{v}=\mathbf{u} \cdot(c \mathbf{v})\)

Carry out the following steps to determine the (smallest) distance between the point \(P\) and the line \(\ell\) through the origin. a. Find any vector \(\mathbf{v}\) in the direction of \(\ell\) b. Find the position vector u corresponding to \(P\). c. Find \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}\). d. Show that \(\mathbf{w}=\mathbf{u}-\) projy \(\mathbf{u}\) is a vector orthogonal to \(\mathbf{v}\) whose length is the distance between \(P\) and the line \(\ell\) e. Find \(\mathbf{w}\) and \(|\mathbf{w}| .\) Explain why \(|\mathbf{w}|\) is the distance between \(P\) and \(\ell\). \(P(1,1,-1) ; \ell\) has the direction of $$\langle-6,8,3\rangle$$.

Find the domains of the following vector-valued functions. $$\mathbf{r}(t)=\frac{2}{t-1} \mathbf{i}+\frac{3}{t+2} \mathbf{j}$$

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 i and \(\mathbf{j}\). What angle does it make with 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 i and \(\mathbf{j}\) ? Explain. e. Find a vector \(\mathbf{v}\) such that \(\alpha=\beta=\gamma .\) What is the angle?

Graph the curve \(\mathbf{r}(t)=\left\langle\frac{1}{2} \sin 2 t, \frac{1}{2}(1-\cos 2 t), \cos t\right\rangle\) and prove that it lies on the surface of a sphere centered at the origin.
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