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Nelson Calculus and Vectors 12

Chris Kirkpatrick, Peter Crippin

Chapter 6

Figure Skating

Educators


Problem 1

State whether each statement is true or false. Justify your decision.
a. If two vectors have the same magnitude, then they are equal.
b. If two vectors are equal, then they have the same magnitude.
c. If two vectors are parallel, then they are either equal or opposite vectors.
d. If two vectors have the same magnitude, then they are either equal or opposite vectors.

Manisha S.
Numerade Educator

Problem 2

For each of the following, state whether the quantity is a scalar or a vector and give a brief explanation why: height, temperature, weight, mass, area, volume, distance, displacement, speed, force, and velocity.

Manisha S.
Numerade Educator

Problem 3

Friction is considered to be a vector because friction can be described as the force of resistance between two surfaces in contact. Give two examples of friction from everyday life, and explain why they can be described as vectors.

Manisha S.
Numerade Educator

Problem 4

Square $A B C D$ is drawn as shown below with the diagonals intersecting at $E$ a. State four pairs of equivalent vectors.
b. State four pairs of opposite vectors.
c. State two pairs of vectors whose magnitudes are equal but whose directions are perpendicular to each other.

Manisha S.
Numerade Educator

Problem 5

5. Given the vector $\overrightarrow{A B}$ as shown, draw a vector
a. equal to $\overrightarrow{A B}$
b. opposite to $\overrightarrow{A B}$
c. whose magnitude equals $|\overrightarrow{A B}|$ but is not equal to $\overrightarrow{A B}$
d. whose magnitude is twice that of $\overrightarrow{A B}$ and in the same direction
e. whose magnitude is half that of $\overrightarrow{A B}$ and in the opposite direction

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Problem 6

Using a scale of $1 \mathrm{cm}$ to represent $10 \mathrm{km} / \mathrm{h}$, draw a velocity vector to represent each of the following:
a. a bicyclist heading due north at $40 \mathrm{km} / \mathrm{h}$
b. a car heading in a southwesterly direction at $60 \mathrm{km} / \mathrm{h}$
c. a car travelling in a northeasterly direction at $100 \mathrm{km} / \mathrm{h}$
d. a boy running in a northwesterly direction at $30 \mathrm{km} / \mathrm{h}$
e. a girl running around a circular track travelling at $15 \mathrm{km} / \mathrm{h}$ heading due east

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Problem 7

The vector shown, $\vec{v}$, represents the velocity of a car heading due north at
$100 \mathrm{km} / \mathrm{h} .$ Give possible interpretations for each of the other vectors shown.

Manisha S.
Numerade Educator

Problem 8

For each of the following vectors, describe the opposite vector.
a. an airplane flies due north at $400 \mathrm{km} / \mathrm{h}$
b. a car travels in a northeasterly direction at $70 \mathrm{km} / \mathrm{h}$
c. a bicyclist pedals in a northwesterly direction at $30 \mathrm{km} / \mathrm{h}$
d. a boat travels due west at $25 \mathrm{km} / \mathrm{h}$

Manisha S.
Numerade Educator

Problem 9

a. Given the square-based prism shown where $A B=3 \mathrm{cm}$ and $A E=8 \mathrm{cm}$ state whether each statement is true or false. Explain. $$\begin{array}{lll}
\text { i) } \overrightarrow{A B}=\overrightarrow{G H} & \text { ii) }|\overrightarrow{E A}|=|\overrightarrow{C G}| & \text { iii) }|\overrightarrow{A D}|=|\overrightarrow{D C}| \quad \text { iv }) \overrightarrow{A H}=\overrightarrow{B G}
\end{array}$$

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Problem 10

James is running around a circular track with a circumference of $1 \mathrm{km}$ at a constant speed of $15 \mathrm{km} / \mathrm{h}$. His velocity vector is represented by a vector tangent to the circle. Velocity vectors are drawn at points $A$ and $C$ as shown. As James changes his position on the track, his velocity vector changes.
a. Explain why James's velocity can be represented by a vector tangent to the circle.
b. What does the length of the vector represent?
c. As he completes a lap running at a constant speed, explain why James's velocity is different at every point on the circle.
d. Determine the point on the circle where James is heading due south.
e. In running his first lap, there is a point at which James is travelling in a northeasterly direction. If he starts at point $A$ how long would it have taken him to get to this point?
f. At the point he has travelled $\frac{3}{8}$ of a lap, in what direction would James
be heading? Assume he starts at point $A$

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Problem 11

$\overrightarrow{A B}$ is a vector whose tail is at (-4,2) and whose head is at (-1,3)
a. Calculate the magnitude of $\overrightarrow{A B}$.
b. Determine the coordinates of point $D$ on vector $\overrightarrow{C D}$, if $C(-6,0)$ and $\overrightarrow{C D}=\overrightarrow{A B}$
c. Determine the coordinates of point $E$ on vector $\overrightarrow{E F}$, if $F(3,-2)$ and $\overrightarrow{E F}=\overrightarrow{A B}$
d. Determine the coordinates of point $G$ on vector $\overrightarrow{G H}$, if $G(3,1)$ and $\overrightarrow{G H}=-\overrightarrow{A B}$

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