9.3.2What can I do with vectors?

Vector Operations

9-88.

Let $\mathbf{a}$ be the vector shown.

Copy $\mathbf{a}$ onto your Section 9.3 Resource Page. Then draw $\mathbf{a}+\mathbf{a}$ . What is another, shorter, name for this new vector?
Draw, with space between them, $2\mathbf{a}$ , $1\mathbf{a}$ , $0\mathbf{a}$ , and $-1\mathbf{a}$ on your resource page. Which of these is equivalent to $\mathbf{0}$ (the zero vector)? Which of these is equivalent to $\mathbf{–a}$ (the opposite vector of $\mathbf{a}$ )?
What is the relationship between the directions of $-\mathbf{a}$ and $\mathbf{a}$ ?
Vector $\mathbf{b}$ indicates a movement of $3$ units to the right and $2$ units up. Draw $\mathbf{b}$ on your resource page.
Now draw $3\mathbf{b}$ and $\frac { 1 } { 2 }\mathbf{b}$ . When working with vectors, the $3$ and the $\frac { 1 } { 2 }$ are scalars. A scalar is a measurement of magnitude (a numerical value) alone.

Grid with ray labeled, a, running 2 & rising 1.

9-89.

VECTOR ADDITION

The diagram at right contains several representations of $\mathbf{c}$ and $\mathbf{d}$ . To draw $\mathbf{c}+\mathbf{d}$ , place the vectors head to tail as shown in the lower part of the grid.

Vectors $\mathbf{c}$ and $\mathbf{d}$ are shown on the resource page. Use diagrams to determine if $\mathbf{c}+\mathbf{d}=\mathbf{d}+\mathbf{c}$ . Then, write an explanation for your conclusion.
Draw and label $\mathbf{c}+\mathbf{d}$ .
Draw and label $\mathbf{c}-\mathbf{d}$ or $\mathbf{c}+-\mathbf{d}$ . Can you draw $\mathbf{c}-\mathbf{d}$ using another method?

Grid with the following: ray labeled, d, running 1 & rising 3, 2 rays, each labeled, c, second one starting at the end of the first one, each running 6 & rising 2, triangle with sides as follows: bottom side labeled, c, running 6 & rising 2, right side labeled, d, running 1 & rising 3, left side labeled, c + d, running 7 & rising 6.

9-90.

THE ARITHMETIC OF VECTORS

Write $\mathbf{c}$ , $\mathbf{d}$ , $\mathbf{c}+\mathbf{d}$ and $-2\mathbf{e}$ in both component and $\mathbf{i, j}$ form.
On the resource page, draw and label $2\mathbf{c}+\mathbf{e}$ .
Write $3\mathbf{d}-2\mathbf{e}$ in $\mathbf{i}, \mathbf{j}$ form without drawing it. Then confirm your answer by drawing the vector.

Grid with ray labeled, e, running left 9 & rising 5, & triangle with sides as follows: bottom side labeled, c, running 6 & rising 2, right side labeled, d, running 1 & rising 3, left side labeled, c + d, running 7 & rising 6.

9-91.

We know that a unit vector is a vector of magnitude $1$ unit.

Copy the vector diagram at right. Draw a unit vector with the same initial point and the same direction as the vector of magnitude $13$ . Then draw and label components for this vector.
Express the unit vector drawn in part (a) in $\mathbf{i}, \mathbf{j}$ form.
Chun was not in class for this problem. Describe a quick way for him to determine the components of a unit vector in the same direction as the vector $a\mathbf{i}+ b\mathbf{j}$ .

right triangle, sides as rays as follows: horizontal leg labeled 5, from left to right, vertical leg labeled 12, from bottom to top, hypotenuse labeled 13, running & rising.

9-92.

Vectors are used to represent physical quantities such as force, which has both magnitude and direction. Suppose you are mowing a lawn on flat, level ground, and are pushing along the handle of the mower with a force of $60$ lbs. (The arrow shows your “push.”) Suppose the handle of the mower makes an angle of $50^\circ$ with the ground.

Draw a vector diagram to illustrate this situation. Label the angle and the force clearly.
Resolve this vector into its horizontal and vertical components.
Which component is doing the useful work of moving the lawn mower ahead?
How much force is actually pushing the lawn mower forward?

Diagram of a lawn mower, heading left, with arrow following the push bar on back, pointing down & left.

9-93.

A boat is headed up a river at an angle of $25^\circ$ at a rate of $20$ ft/sec. The speed of the current is $5$ ft/sec. The river is $100$ feet wide.

Express the velocity of the boat and the velocity of the current, each as a vector in component form.
The actual motion of the boat is the combination of the motion of the boat and the current in the river. Write a vector that gives the actual motion of the boat.
How long will it take the boat to cross the river? Which component of the vector from part (b) did you use to determine your answer?
How far up or down river will the boat arrive?

9-94.

Thoroughly investigate the graph of $f(x) = e^{−x^2}$ . Identify all of the important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify point(s) of inflection, extrema, and intercepts, and provide graphs of $y = f ^\prime(x)$ and $y = f^{\prime\prime}(x)$ . Be sure to justify all statements both graphically and analytically. Homework Help ✎

9-95.

Use Euler’s Method to sketch a solution for $\frac { d y } { d x }= -y$ starting at $(–2, –3)$ with $∆x = 0.5$ . Homework Help ✎

9-96.

Approximate the area under $f(x) = 2 \cos\sqrt { x }+3$ for $0 ≤ x ≤ 3$ using six midpoint rectangles of equal width. Homework Help ✎

9-97.

Multiple Choice: The function f given below has discontinuities at: Homework Help ✎ $f ( x ) = \left\{ \begin{array} { l l } { - ( 2 x + 3 ) ^ { 2 } } & { \text { for } x < - 1 } \\ { x ^ { - 2 / 3 } + 2 x } & { \text { for } - 1 \leq x \leq 1 } \\ { 3 \operatorname { cos } ( 0.1 x ) } & { \text { for } x > 1 } \end{array} \right.$

$x = 0$

$x = -1, 0$

$x = -1, 1$

$x = 0, 1$

$x = 1$

9-98.

Consider each of the infinite series below. For each series, decide if it converges or diverges and justify your conclusion. If the series converges, calculate its sum. Homework Help ✎

$4 + 4 + 4 + 4 + …$

$\frac { 1 } { 10 } + \frac { 1 } { 100 } + \frac { 1 } { 1000 } + \ldots$

$10 + 9 + 8 + 7 + …$

$–2 +\frac { 6 } { 5 } - \frac { 18 } { 25 } + \frac { 54 } { 125 } - \ldots$

9-99.

For the following sets of parametric equations, express $y$ as a function of $x$ . Then compare the graphs of each parametric curve with its rectangular curve. Cite any similarities and any differences. Homework Help ✎

$x = \tan(t)$ and $y = \tan^2(t)$

$x = \log(t)$ and $y = 1 + t^2$

9-100.

If $\mathbf{z} = -4\mathbf{i} - 6\mathbf{j}$ , what are $|| z ||$ and the direction? Homework Help ✎

9-101.

Baby Mathilde loves milk. The function $M$ represents the rate that Mathilde drinks milk in ounces per hour from when she wakes up at 6:15 a.m., $t=0$ , to the time she takes her midday nap. Homework Help ✎

Given the situation described above, interpret the meaning of $\int_0^7M(t)dt=32$ .
Write a complete description about what the integral is computing. Use correct units and be sure to include the bounds in your description.
Compute the value of $\frac { 1 } { 7 } \int _ { 0 } ^ { 7 } M ( t ) d t$ and interpret its meaning in the context of this problem.

9-102.

Locate vectors $\mathbf{f}$ and $\mathbf{g}$ on the Section 9.3 Resource Page. Homework Help ✎

Write $\mathbf{f}$ and $\mathbf{g}$ in $\mathbf{i}, \mathbf{j}$ form. Then write the $\mathbf{i}, \mathbf{j}$ form for $2\mathbf{f}-\frac { 1 } { 3 }\mathbf{g}$ .
Draw and label $2\mathbf{f}-\frac { 1 } { 3 }\mathbf{g}$ . Does your diagram confirm your answer from part (a)?

9-103.

On the resource page, draw $\mathbf{c-d}$ and $\mathbf{d-c}$ . Homework Help ✎

What one word describes the relationship between $\mathbf{c-d}$ and $\mathbf{d-c}$ ?
Draw $\mathbf{f}$ and $\mathbf{g}$ with a third vector from the head of $\mathbf{f}$ to the head of $\mathbf{g}$ as shown at right. Label the third vector using $\mathbf{f}$ and $\mathbf{g}$ .

Grid with 3 rays forming a triangle, left top side labeled, f, runs 4 & up 2, from left vertex to top right vertex, left bottom side labeled, g, runs 3 & falls 6, from left vertex to bottom right vertex, right side runs left 1 & falls 8, from top right vertex to bottom right vertex.

Calculus Third Edition