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 \(Q\text{.}\))
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 \(\vec{d} = \vec{a} - \vec{b}\text{.}\)
Figure1.9.2.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?
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 \(\vec{d} = \vec{a} - \vec{b}\text{.}\)
Figure1.9.3.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?
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.
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?
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*S1.9.6.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?
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!
SubsectionApply
Activity1.9.7.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):
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.
