9.3.1What are vectors?

Introduction to Vectors

When we describe physical quantities such as velocity and acceleration, it is useful to think of them as vectors. Vectors are a convenient tool used in physics and calculus to describe physical phenomenon that have force and direction, such as wind or river currents.

9-72.

Town B is $5$ miles from Town A.
Town C is $2$ miles from Town B.
How far is town C from Town A?
Is there more than one possible answer?

Segment with 2 points, left point labeled Town A, right point labeled Town B, segment between points is bolded.

9-73.

It turns out that Town B is directly east of Town A, and Town C is directly north of Town B.

How far is Town C from Town A?
Good news! Funds have just been approved for a new highway from Town A to Town C. Assuming the terrain is flat and there is nothing to prevent the highway from being perfectly straight, tell the engineers what direction the highway should take by determining a suitable angle.
Straight highways already connect Town A with Town B and Town B with Town C. The diagram at right illustrates two different ways to travel from Town A to Town C. Draw a similar diagram showing two different ways to travel from Town C to Town B along these highways.

9-74.

Obtain a Section 9.3 Resource Page.

Town F is $8$ miles west and $6$ miles north of Town E. Draw an arrow that accurately shows the direct route from Town E to Town F. Label it with the distance between the two towns and an angle.
Yonitte bikes from Town F to Town G, $2$ miles east and $5$ miles south. Draw a single arrow to represent her trip. Label the arrow with the distance Yonitte traveled.

Both the arrow showing the direct route from Town E to Town F and that representing Yonitte’s trip are vectors. Vectors have both length (magnitude) and direction.
Suppose Dennis takes a trip from Town E to Town F, and then to Town G. Trace his path on your resource page. How far does Dennis travel?
Now draw a vector from Town E to Town G, representing Dennis’ starting and ending points. This vector is called the resultant vector (or, in this case, the displacement vector) of Dennis’ trip. What was Dennis’ total displacement?

9-75.

Vectors are also useful for representing forces.

While digging a hole to plant a new pear tree, Alfonso pushes on a shovel with a force of $60$ pounds at an angle of $67^\circ$ with the ground. Draw a vector to represent the force Alfonso applies to the shovel. Label $60$ pounds and $67^\circ$ appropriately in your diagram.
Although it makes sense to say that the length of a trip from Town A to Town B is $5$ miles, it makes no sense to say that the length of Alfonso’s force is $60$ pounds. Instead, we say the magnitude of Alfonso’s force is $60$ pounds. The force Alfonso applies to the shovel can be represented by either by a single vector, or by a vertical vector followed by a horizontal vector (or vice versa). The second representation breaks his force into horizontal and vertical parts, called components. Determine the magnitude of the vertical component of Alfonso’s force.

9-76.

Two attributes are needed to specify motion or force: magnitude (length) and direction. A single arrow, called a vector, can represent both attributes at once. Vectors are the best way to represent physical quantities that have both magnitude and direction, such as the motion of an object.

Which of the common quantities listed below are best represented by vectors?

length

force

geometric reflection

geometric translation

time

cost

speed (like $60$ mph)

velocity

weight

mass

acceleration

direction

Math Notes

Vector Notations

A vector is a quantity, usually represented by an arrow, having magnitude (or length) and direction.

A vector can be named using its initial and end points, or it can be given a single letter name. A vector that is defined by its endpoints will be written in italicized capital letters with an arrow over the top. For example, the vector that begins at point $A$ and ends at point $B$ will be written as $\overrightarrow { A B }$ . In handwritten work, it is impossible to indicate a vector using bold text. To solve this problem, mathematicians often indicate a vector quantity by placing an arrow over the letter representing the vector. Thus, $\mathbf{v}$ and $\overrightarrow { V }$ represent the same vector, but the first is used in typewritten work and the second in handwritten work.

Vectors can also be written in component form. This represents the horizontal and vertical directions of the vector. In the diagram above, $\overrightarrow { A B }=\langle 3,2 \rangle$ because it indicates a movement is that is $3$ units to the right and $2$ units up. Vector $\mathbf{v}=\langle 2 , - 2 \rangle$ . We will use the notation $\langle a , b \rangle$ to represent a vector in component form. Some books write component form as $\left(a, b\right)$ just like ordered pairs.

9-77.

Vectors $\mathbf{p}$ and $\mathbf{q}$ , shown below, are equivalent vectors. Equivalent vectors are vectors that have the same magnitude and direction.

4 rays as follows: first labeled, p, runs 5 & falls 2, second labeled, q, runs 5 & falls 2, third labeled, s, runs 7 & rises 2, fourth labeled, r, runs 7 & rises 2. . 5 rays as follows: first labeled, j, runs left 6 & falls 1, second labeled, h, runs left 12 & falls 2, third labeled, l, rises up 4, fourth labeled, k, rises up 4, fifth labeled, m, falls down 4.

Name two other pairs of equivalent vectors.
On your resource page, draw a vector equivalent to $\mathbf{d}$ . Compare your vector with the vectors your teammates drew. Are all of your vectors equivalent?

9-78.

In the diagram at right, $ABCD$ is a parallelogram. Write another name for each of the vectors described below using only $\mathbf{u}, \mathbf{v}, \mathbf{w}$ , and $\mathbf{x}$ . Do not add anything to the diagram. All are possible.

$\overrightarrow { D C }$

$\overrightarrow { B C }$

$\overrightarrow { A B }$

$\overrightarrow { A B } + \overrightarrow { B C }$

$\mathbf{u + v}$

$\mathbf{v + w}$

$\mathbf{w + v}$

$-\overrightarrow { A B }$

$\overrightarrow { B D }$

parallelogram, A B C D, sides & diagonals, labeled as follows: side A to D, labeled v, side A to B, labeled u, diagonal A to C, labeled x, diagonal d to b, labeled w.

Math Notes

Describing Vectors

3 rays, first labeled I, running 1 unit right, second labeled, j, running 1 unit up, third labeled, negative 3 I + 4 J, running 3 left & 4 up. The vector $\mathbf{i}=\langle 1,0 \rangle$ is a horizontal vector with a length of $1$ unit. The vector $\mathbf{j}=\langle 0,1 \rangle$ is a vertical vector, also with a length of $1$ unit. Any vector can be written in $\mathbf{i, j}$ form. For example, a vector in component form such as $\langle - 3,4 \rangle$ can be written as $–3\mathbf{i} + 4\mathbf{j}$ .

The magnitude of a vector is a scalar quantity and can be calculated by using the Pythagorean Theorem. Given vector $\mathbf{v}=\langle a , b \rangle$ , the magnitude of $\mathbf{v}$ (written $| V | |$ ) is $| | v | | = \sqrt { a ^ { 2 } + b ^ { 2 } }$ .

The direction of a vector is the counterclockwise angle that the vector makes with a horizontal ray pointing to the right. This can be calculated using trigonometry. Given vector $\mathbf{v}=\langle a , b \rangle$ , the direction is $\theta = \operatorname { tan } ^ { - 1 } ( \frac { b } { a } )$ .

A unit vector is a vector with a magnitude of $1$ unit. To find a unit vector with the same direction of a given vector, each component must be divided by the magnitude. In this case, the unit vector in the same direction as $\langle - 3,4 \rangle$ is $\langle - \frac { 3 } { 5 } , \frac { 4 } { 5 } \rangle$ .

9-79.

A vector has an initial point of $(–2, 4)$ and an end point of $(5, –4)$ . Note: Refer to the Math Notes boxes in this lesson for examples. Homework Help ✎

Draw the vector on your resource page and write it in both component and $\mathbf{i, j}$ form.
Calculate the magnitude and direction of the vector.

9-80.

Write the integral that calculates the volume generated when the region bounded by $y = x^2$ and $x = y^4$ is rotated: Homework Help ✎

About the $x$ -axis.
About the $y$ -axis.
About the line $x = 2$ .

9-81.

Multiple Choice: The arc length of the curve $y=2\sin\sqrt{x}$ over the interval $1 ≤ x ≤ π^2$ is closest to: Homework Help ✎

$1.4$

$2.3$

$8.9$

$9.2$

$9.6$

9-82.

$\lim \limits _ { h \rightarrow 0 } \frac { \operatorname { tan } ^ { - 1 } ( x + h ) - \operatorname { tan } ^ { - 1 } ( x ) } { h }$ is equivalent to: Homework Help ✎

$\frac { 1 } { 1 + x ^ { 2 } }$

$\frac { 1 } { x \sqrt { x ^ { 2 } - 1 } }$

$\tan(x)$

$\frac { 1 } { x \sqrt { 1 - x ^ { 2 } } }$

$\sec^2(x)$

9-83.

Multiple Choice: A projectile is launched vertically from the ground with an initial velocity of $50$ feet per second. Which of the following is closest to its maximum height? Homework Help ✎

$32$ ft

$39$ ft

$46$ ft

$50$ ft

$128$ ft

9-84.

Determine values of $k$ and $m$ such that $f$ , given below, is differentiable for all $x$ . Homework Help ✎

$f ( x ) = \left\{ \begin{array} { l l } { k x ^ { 2 } - 3 x + 2 } & { \text { for } x < 2 } \\ { 5 x + m } & { \text { for } x \geq 2 } \end{array} \right.$

9-85.

Convert each of the following sets of parametric equations to rectangular form. Homework Help ✎

$x = \sin^2(t)$ and $y =\sqrt [ 3 ] { \operatorname { sin } ( t ) }$

$x = t^{12}$

9-86.

$\overrightarrow { A B }$ is the correct notation for either a ray or a vector. Compare and contrast ray $\overrightarrow { A B }$ with vector $\overrightarrow { A B }$ . How are they the same; how are they different? Include diagrams in your answer. Homework Help ✎

9-87.

Write an infinite geometric series with a sum of $3$ . Homework Help ✎

Calculus Third Edition

Vector Notations

Describing Vectors