3.1.E: Problems on Vectors in \(E^{n}\) (Exercises)
Prove by induction on \(n\) that
\[
\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\left(y_{1}, y_{2}, \ldots, y_{n}\right) \text { iff } x_{k}=y_{k}, k=1,2, \ldots, n.
\]
[Hint: Use Problem 6(ii) of Chapter 1, §§1-3, and Example (i) in Chapter 2, §§5-6.]
Complete the proofs of Theorems 1 and 3 and Notes 3 and 8.
Given \(\overline{x}=(-1,2,0,-7), \overline{y}=(0,0,-1,-2),\) and \(\overline{z}=(2,4,-3,-3)\) in \(E^{4},\) express \(\overline{x}, \overline{y},\) and \(\overline{z}\) as linear combinations of the basic unit vectors. Also, compute their absolute values, their inverses, as well as their mutual sums, differences, dot products, and distances. Are any of them orthogonal? Parallel?
With \(\overline{x}, \overline{y},\) and \(\overline{z}\) as in Problem \(3,\) find scalars \(a, b,\) and \(c\) such that
\[
a \overline{x}+b \overline{y}+c \overline{z}=\overline{u},
\]
when
\[
\begin{array}{rlrl}{(\mathrm{i}) \overline{u}} & {=\overline{e}_{1} ;} & {} & {\text { (ii) } \overline{u}=\overline{e}_{3}}; \\ {\text { (iii) } \overline{u}} & {=(-2,4,0,1) ;} & {} & {\text { (iv) } \overline{u}=\overline{0}}.\end{array}
\]
A finite set of vectors \(\overline{x}, \overline{x}_{2}, \ldots, \overline{x}_{m}\) is said to be dependent iff there are scalars \(a_{1}, \ldots, a_{m},\) not all zero, such that
\[
\sum_{k=1}^{m} a_{k} \overline{x}_{k}=\overline{0},
\]
and independent otherwise. Prove the independence of the following sets of vectors:
(a) \(\overline{e}_{1}, \overline{e}_{2}, \ldots, \overline{e}_{n}\) in \(E^{n}\);
(b) \((1,2,-3,4)\) and \((2,3,0,0)\) in \(E^{4} ;\)
(c) \((2,0,0),(4,-1,3),\) and \((0,4,1)\) in \(E^{3} ;\)
(d) the vectors \(\overline{x}, \overline{y},\) and \(\overline{z}\) of Problem 3.
Prove (for \(E^{2}\) and \(E^{3} )\) that
\[
\overline{x} \cdot \overline{y}=|\overline{x}||\overline{y}| \cos \alpha ,
\]
where \(\alpha\) is the angle between the vectors \(\overrightarrow{0 x}\) and \(\overrightarrow{0 y} ;\) we denote \(\alpha\) by \(\langle\overline{x}, \overline{y}\rangle\).
[Hint: Consider the triangle \(\overline{0} \overline{x} \overline{y},\) with sides \(\overline{x}=\overrightarrow{0 x}, \overline{y}=\overrightarrow{0 y},\) and \(\overrightarrow{x y}=\vec{y}-\vec{x}\) (see Definition 7 ). By the law of cosines,
\[
|\vec{x}|^{2}+|\vec{y}|^{2}-2|\vec{x}||\vec{y}| \cos \alpha=|\vec{y}-\vec{x}|^{2}.
\]
Now substitute \(|\vec{x}|^{2}=\vec{x} \cdot \vec{x},|\vec{y}|^{2}=\vec{y} \cdot \vec{y},\) and
\[
|\vec{y}-\vec{x}|^{2}=(\vec{y}-\vec{x}) \cdot(\vec{y}-\vec{x})=\vec{y} \cdot \vec{y}+\vec{x} \cdot \vec{x}-2 \vec{x} \cdot \vec{y} .(\mathrm{Why} ?)
\]
Then simplify.]
Motivated by Problem \(6,\) define in \(E^{n}\)
\[
\langle\overline{x}, \overline{y}\rangle=\arccos \frac{\overline{x} \cdot \overline{y}}{|\overline{x}||\overline{y}|} \text { if } \overline{x} \text { and } \overline{y} \text { are nonzero. }
\]
(Why does an angle with such a cosine exist?) Prove that
(i) \(\overline{x} \perp \overline{y}\) iff \(\cos \langle\overline{x}, \overline{y}\rangle= 0,\) i.e., \(\langle\overline{x}, \overline{y}\rangle=\frac{\pi}{2}\);
(ii) \(\sum_{k=1}^{n} \cos ^{2}\left\langle\overline{x}, \overline{e}_{k}\right\rangle= 1\).
Continuing Problems 3 and \(7,\) find the cosines of the angles between the sides, \(\overrightarrow{x y}, \quad \overrightarrow{y z},\) and \(\overrightarrow{z x}\) of the triangle \(\overline{x} \overline{y} \overline{z},\) with \(\overline{x}, \overline{y},\) and \(\overline{z}\) as in Problem 3.
Find a unit vector in \(E^{4},\) with positive components, that forms equal angles with the axes, i.e., with the basic unit vectors (see Problem 7).
Prove for \(E^{n}\) that if \(\overline{u}\) is orthogonal to each of the basic unit vectors \(\overline{e}_{1}\), \(\overline{e}_{2}, \ldots, \overline{e}_{n},\) then \(\overline{u}=\overline{0} .\) Deduce that
\[
\overline{u}=\overline{0} \text { iff }\left(\forall \overline{x} \in E^{n}\right) \overline{x} \cdot \overline{u}=0.
\]
Prove that \(\overline{x}\) and \(\overline{y}\) are parallel iff
\[
\frac{x_{1}}{y_{1}}=\frac{x_{2}}{y_{2}}=\cdots=\frac{x_{n}}{y_{n}}=c \quad\left(c \in E^{1}\right),
\]
where \(" x_{k} / y_{k}="c\) is to be replaced by \(" x_{k}=0 "\) if \(y_{k}=0\).
Use induction on \(n\) to prove the Lagrange identity (valid in any field),
\[
\left(\sum_{k=1}^{n} x_{k}^{2}\right)\left(\sum_{k=1}^{n} y_{k}^{2}\right)-\left(\sum_{k=1}^{n} x_{k} y_{k}\right)^{2}=\sum_{1 \leq i<k \leq n}\left(x_{i} y_{k}-x_{k} y_{i}\right)^{2}.
\]
Hence find a new proof of Theorem 4\(\left(\mathrm{c}^{\prime}\right)\).
Use Problem 7 and Theorem 4\(\left(\mathrm{c}^{\prime}\right)(\text { "equality") to show that two nonzero }\) vectors \(\overline{x}\) and \(\overline{y}\) in \(E^{n}\) are parallel iff \(\cos \langle\overline{x}, \overline{y}\rangle=\pm 1\).
(i) Prove that \(|\overline{x}+\overline{y}|=|\overline{x}|+|\overline{y}|+|\overline{y}|\) iff \(\overline{x}=t \overline{y}\) or \(\overline{y}=t \overline{x}\) for some \(t \geq 0\); equivalently, iff \(\cos \langle\overline{x}, \overline{y}\rangle= 1\) (see Problem 7\() .\)
(ii) Find similar conditions for \(|\overline{x}-\overline{y}|=|\overline{x}|+|\overline{y}|\).
[Hint: Look at the proof of Theorem 4\(\left(\mathrm{d}^{\prime}\right) . ]\)