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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 1.9 Practice, Study, and Apply - Vectors

Subsection Practice

Calculation 1.9.1. Sums and Differences of Vectors.

Use the figure to determine a single vector that is equivalent to the given summation of vectors in each of the cases below.  (The notation \(\vec{PQ}\) represents a vector pointing from point \(P\) to point \(Q\text{.}\))
Figure 1.9.1. Two vectors.
  1. \(\displaystyle \vec{PQ} + \vec{QR}\)
  2. \(\displaystyle \vec{RP} + \vec{PS}\)
  3. \(\displaystyle \vec{QS} + \vec{PS}\)
  4. \(\displaystyle \vec{RS} + \vec{SP} + \vec{PQ}\)
Answer.
  1. \(\displaystyle \vec{PR}\)
  2. \(\displaystyle \vec{RS}\)
  3. \(\displaystyle \vec{QP}\)
  4. \(\displaystyle \vec{RQ}\)

Calculation 1.9.2. Two Vectors I.

In the following figure, the magnitudes of the vectors are \(|\vec{a}| = 8\) and \(|\vec{b}| = 4.5\text{.}\) Assume that \(\vec{c} = \vec{a} + \vec{b}\) and \(\vec{d} = \vec{a} - \vec{b}\text{.}\)
Figure 1.9.2. Two vectors.
Determine the magnitude of the vectors \(\vec{c}\) and \(\vec{d}\text{?}\) What is the angle to each vector from the positive \(x\)-axis?
Answer.
\(\vec{c}\text{:}\) \(3\) at \(109.6^o\)
\(\vec{d}\text{:}\) \(8.1\) at \(167.5^o\)

Calculation 1.9.3. Two Vectors II.

In the following figure, the magnitudes of the vectors are \(|\vec{a}| = 5\) and \(|\vec{b}| = 5\text{.}\) Assume that \(\vec{c} = \vec{a} + \vec{b}\) and \(\vec{d} = \vec{a} - \vec{b}\text{.}\)
Figure 1.9.3. Two vectors.
Determine the magnitude of the vectors \(\vec{c}\) and \(\vec{d}\text{?}\) What is the angle to each vector from the positive \(x\)-axis?
Answer.
\(\vec{c}\text{:}\) \(4\) at \(-42.5^o\)
\(\vec{d}\text{:}\) \(8.87\) at \(227.5^o\)

Calculation 1.9.4. Vector Contest.

Three vectors add together to equal \(0\text{.}\)  One vector has magnitude \(3\) and points in the positive \(x\)-direction; a second vector has magnitude \(5\) and points at \(120^o\) from the positive \(x\)-axis. Determine the third vector as a magnitude and direction.
Answer.
\(4.4\) at \(-97^o\)

Subsection Study

Explanation 1.9.5. Subtracting Equal Vectors.

In each case below, the top vector is being subtracted from the bottom vector. All of the vectors have the same magnitude. Is the magnitude of the difference in the left case greater than, less than, or equal to the magnitude of the difference in the right case?
Figure 1.9.4. Two pairs of vectors.
Solution.
One way to subtract two vectors is to place the vectors tail-to-tail and then draw the vector that points from the tip of one vector to the tail of the other. In this case, the difference in the left case is clearly less than the difference in the right case.

A*R*C*S 1.9.6. The Large Tree.

You and a friend are each lost in the same forest. They message you that they are directly northeast of a very large tree that is about \(3\) miles away. You are about \(2\) miles away from the same large tree, but you are \(25^o\) west of south of it. About how far are you from your friend? What direction would you need to walk to reach your friend?
Tip.
Remember to solve problems symbolically first! It can help to choose sensible variable names for all numbers given in the problem, and for any numbers that you want to find.
When making sense of your symbolic answer, a good starting point is to check which variables your answer does and does not depend on. As part of making sense, always make sure to discuss not just what your answer says, but what the answer should say and why!

Subsection Apply

Activity 1.9.7. The Trees upon the Hillside.

A surveyor measures a hillside and describes the height of the hill \(y\) (in meters) as the following function of distance from the base of the hill \(x\) (also measured in meters):
\begin{equation*} y(x) = a_1x^3 - a_2x^5 \end{equation*}
He reports that \(a_1 = 1.5 \times 10^{-5}\) and \(a_2 = 1.0 \times 10^{-10}\text{.}\) He has also identified three especially large trees (about \(30 \mathrm{~m}\) tall) whose bases are located at \(x_1 = 5 \mathrm{~m}\text{,}\) \(x_2 = 257 \mathrm{~m}\text{,}\) and \(x_3 = 300 \mathrm{~m}\text{.}\)
(a) Sensemake.
It looks like the surveyor forgot to report the specific units for the constants \(a_1\) and \(a_2\text{.}\) Determine appropriate units for these constants. Describe why these are sensible units given the equation for \(y(x)\text{.}\)
(b) Represent.
Sketch and label a quantitatively accurate diagram of the hillside and the trees.
Tip.
Make sure to identify your origin, label your axes, and identify any relevant points of interest.
(c) Calculate.
Determine the displacement vector from the base of tree 2 to the base of tree 3.
Tip.
Remember to do all calculations symbolically before you plug in numbers, following the steps in Figure 2.2.4.

Activity 1.9.8. Math Review.

(a)
Consider the integral \(V = \int_{2}^{7} x^2 dx\text{.}\)
1. Describe the meaning of each symbol and number in this expression.
2. Determine the value of V.
3. Sketch a graph that gives meaning to this integral.
4. If x is measured in meters, what are the units of V?
(b)
Use an integral to determine the area of the triangle shown below. What is the meaning of the infinitesimal in the integral you use?
Figure 1.9.5.
(c)
Use an integral to determine the area of the shape shown below, where the curve is a parabola with its vertex at the lower left corner.
Figure 1.9.6.

References References

[1]
Practice activities provided by BoxSand: https://boxsand.physics.oregonstate.edu/welcome.
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