1.
\(\u+\v\)
Section 2.2 Vector Basics: In-Class Practice
Worksheet 2.2.1 Worksheet
Exercise Group.
Let \(\u=\left(\begin{array}{r} -4\\8\\1 \end{array}\right)\) and \(\v=\left(\begin{array}{r} 4\\0\\-3\end{array}\right)\text{.}\) Determine the following:
2.
\(\u-\v\)
3.
\(2\u+3\v\)
4.
\(\|\v\|\)
5.
\(\|2\v\|\)
6.
\(\|2\u+3\v\|\) (Is \(\|2\u+3\v\|=2\|\u\|+3\|\v\|?)\)
7.
Graph \(\left(\begin{array}{c}\cos\theta\\\sin\theta\end{array}\right)\) for \(\displaystyle{\theta=0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2}}.\) Use a straightedge and label the axes and the \(x,y-\)coordinates of the terminal point of each vector.
8.
Prove that for any \(\v\ne\vec{0},\) the vector \(\displaystyle{\frac{\v}{\|\v\|}}\) is parallel to \(\v.\)
Exercise Group.
Let \(\u=\left(\begin{array}{r} 2\\-2\\1 \end{array} \right)\in\R^3.\)
9.
Find the norm \(\|\u\|\) of \(\u.\)
10.
Find two vectors \(\v_1\) and \(\v_2\) parallel to \(\u.\) Be sure to check \(\u \parallel \v_1\) and \(\u \parallel \v_2.\)
11.
Find a unit vector \(\widehat{\boldsymbol{u}}\) parallel to (or, if you prefer, “in the direction of”) \(\u.\) Be sure to verify that your candidate has the desired properties, citing any relevant definitions. Can you find another one? Explain.
12.
Graph \(\left(\begin{array}{c}3\\4\end{array}\right)\) and the unit vector parallel to it, labeling the coordinates of the terminal point for each. Label your axes.
13.
Find the unit vector parallel to \(\left(\begin{array}{c}3\\4\\12\end{array}\right),\) and find the vector of length \(4\) parallel to \(\left(\begin{array}{c}3\\4\\12\end{array}\right).\)
14.
Let \(\u=\begin{pmatrix}u_1\\u_2\\\vdots\\u_n \end{pmatrix}\in\R^n.\) Find a vector \(\v=\begin{pmatrix}v_1\\ v_2\\\vdots\\ v_n\end{pmatrix}\in\R^n\) for which \(3\u+5\v=\vec{0}\text{.}\)
15.
Find a linear combination \(a\left(\begin{array}{r} 2\\-3 \end{array} \right)+b\left(\begin{array}{r}-3\\4 \end{array} \right)\) which equals \(\left(\begin{array}{r}-3\\2 \end{array} \right)\text{.}\)
Hint.
To do this carefully, write \(a\left(\begin{array}{r} 2\\-3 \end{array} \right)+b\left(\begin{array}{r}-3\\4 \end{array} \right)=\left(\begin{array}{r}-3\\2 \end{array} \right)\) as a \(2\times 2\) system and use algebra to find the coefficients \(a,b\in\R\text{.}\)
16.
Find a linear combination of \(\left(\begin{array}{r} 2\\-3 \end{array} \right)\) and \(\left(\begin{array}{r}-3\\4 \end{array} \right)\) which equals \(\begin{pmatrix}{0}\\0 \end{pmatrix} \text{.}\)
17.
Find two nonzero linear combinations of \(\left(\begin{array}{r} 2\\-3 \end{array} \right)\) and \(\left(\begin{array}{r}4\\-6 \end{array} \right)\) which equal \(\begin{pmatrix}{0}\\0 \end{pmatrix} \text{.}\) Could you find more such linear combinations?
18.
Find a linear combination of \(\left(\begin{array}{r} 2\\-3\\1 \end{array} \right)\) and \(\left(\begin{array}{r}-3\\4\\2 \end{array} \right)\) which equals \(\left(\begin{array}{r}-3\\2\\3 \end{array} \right)\) or explain why no such solution exists.
Hint.
Your work on Problem Worksheet Exercise 2.2.1.15 might be helpful.
19.
Find the unit vector in the direction of the ray making an angle of \(\displaystyle{\frac{4\pi}{3}}\) radians counterclockwise from the positive \(x-\)axis in \(\R^2.\)
20.
Prove Theorem 2.1.21.
21.
Prove Theorem 2.1.26.
