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Elementary Linear Algebra: An Active Learning Approach

Section 2.6 Vector Properties: Homework

Exercises Exercises

Exercise Group.

For each pair of vectors, verify both the Cauchy-Schwarz inequality and the triangle inequality, and find the (radian measure of the) angle between the two vectors.
1.
\(\u=\left(\begin{array}{r}1\\3\\-2 \end{array} \right),\,\v=\left(\begin{array}{r}1\\3\\-1 \end{array} \right)\)
2.
\(\u=\left(\begin{array}{r}2\\4\\-5 \end{array} \right),\,\v=\left(\begin{array}{r}2\\4\\-4 \end{array} \right)\)
3.
\(\u=\left(\begin{array}{r}1\\2\\3 \end{array} \right),\,\v=\left(\begin{array}{r}-1\\-2\\-3 \end{array} \right)\)
4.
\(\u=\left(\begin{array}{r}3\\4\\5 \end{array} \right),\,\v=\left(\begin{array}{r}-3\\-4\\-5 \end{array} \right)\)
5.
\(\u=\left(\begin{array}{r}5\\2\\-1 \end{array} \right),\,\v=\left(\begin{array}{r}-2\\0\\-10 \end{array} \right)\)
6.
\(\u=\left(\begin{array}{r}2\\3\\-1 \end{array} \right),\,\v=\left(\begin{array}{r}-1\\2\\4 \end{array} \right)\)

7.

Find a one-dimensional infinite set of vectors orthogonal to \(\v=\left(\begin{array}{r} 3\\2\\-1 \end{array} \right)\) and express this set in element-form notation.

8.

Find a one-dimensional infinite set of vectors orthogonal to \(\left(\begin{array}{r} 3\\2\\-1 \end{array} \right)\) distinct from the set you found for Exercise 2.6.7 and write this set expression in element-form notation.

9.

Find a two-dimensional infinite set of vectors orthogonal to \(\v=\left(\begin{array}{r} -2\\6\\5 \end{array} \right)\) and express this set in element-form notation.

10.

Find a two-dimensional infinite set of vectors orthogonal to \(\v=\left(\begin{array}{r} 3\\-1\\8 \end{array} \right)\) and express this set in element-form notation.

11.

Prove that an arbitrary element \(\vec{v}=\begin{pmatrix}v_1\\v_2\\v_3 \end{pmatrix}\) in the set you found in Exercise 2.6.10 is indeed orthogonal to \(\left(\begin{array}{r} 3\\-1\\8 \end{array} \right)\) by showing that the dot product \(3v_1-v_2+8v_3=0.\)

15. Equality in the triangle inequality.

Let \(\u,\v\in\R^n.\) Prove that
\begin{equation*} \|\u+\v\|=\|\u\|+\|\v\| \end{equation*}
if and only if \(\u\parallel\v\text{.}\)

16.

Find nonzero vectors \(\u,\v,\w\) which are all orthogonal to \(\left(\begin{array}{r} 1\\1\\-1\\1 \end{array} \right)\) and orthogonal to each other as well.

17.

Find three mutually orthogonal (all pairs are orthogonal) vectors in \(\R^3.\) At least one of your vectors must contain no zeros.

18.

Find two unit vectors orthogonal to \(\left(\begin{array}{r}6\\-1 \end{array} \right).\)

19.

Find five unit vectors orthogonal to \(\left(\begin{array}{r} 1\\-1\\2\\1\\-2\\1 \end{array} \right).\) No pair of your proposed vectors may be parallel or anti-parallel. In each case, be sure to verify orthogonality, citing Definition 2.4.6, and verify that the length equals 1, citing Definition 2.1.20, for each proposed unit vector.

Exercise Group.

True or false, with proof. If true, prove the statement. If false, provide a counterexample.
20.
Parallelity is transitive: If \(\u,\v,\w\in\R^n\) satisfy \(\u\parallel\v\) and \(\v\parallel\w,\) then \(\u\parallel\w\text{.}\)
21.
Orthogonality is transitive: If \(\u,\v,\w\in\R^n\) satisfy \(\u\perp\v\) and \(\v\perp\w,\) then \(\u\perp\w.\)
22.
If two vectors in \(\R^n\) are orthogonal to the same vector, then those vectors must be parallel. That is, if \(\vec{a}\perp\v\) and \(\vec{b}\perp\v\) then \(\vec{a}\parallel\vec{b}.\)
23.
If two vectors \(\u,\v\) in \(\R^2\) are orthogonal to the same vector in \(\R^2,\) then \(\u\) and \(\v\) must be parallel.
24.
If two vectors \(\u,\v\) in \(\R^2\) are orthogonal to the same nonzero vector in \(\R^2,\) then \(\u\) and \(\v\) must be parallel.
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