# A line L through the origin in RR^3 can be represented by parametric equations of the form x = at, y = bt, and z = ct.

Question
Alternate coordinate systems

A line L through the origin in $$\displaystyle\mathbb{R}^{{3}}$$ can be represented by parametric equations of the form x = at, y = bt, and z = ct. Use these equations to show that L is a subspase of $$RR^3$$  by showing that if $$v_1=(x_1,y_1,z_1)\ and\ v_2=(x_2,y_2,z_2)$$  are points on L and k is any real number, then $$kv_1\ and\ v_1+v_2$$  are also points on L.

2021-01-31
Let $$\displaystyle{v}_{{1}}={\left({x}_{{1}},{y}_{{1}},{z}_{{1}}\right)}{\quad\text{and}\quad}{v}_{{2}}={\left({x}_{{2}},{y}_{{2}},{z}_{{2}}\right)}$$ be elements of L.Then
$$\displaystyle{\left\lbrace\begin{array}{c} {x}_{{1}}={a}{t}_{{1}}\\{y}_{{1}}={b}{t}_{{1}}\\{z}_{{1}}={c}{t}_{{1}}\end{array}\right.}{\quad\text{and}\quad}{\left\lbrace\begin{array}{c} {x}_{{2}}={a}{t}_{{2}}\\{y}_{{2}}={b}{t}_{{2}}\\{z}_{{2}}={c}{t}_{{2}}\end{array}\right.}$$
for some real numbers $$\displaystyle{t}_{{1}}{\quad\text{and}\quad}{t}_{{2}}$$.
let $$\displaystyle{w}={v}_{{1}}+{v}_{{2}}={\left({x}_{{1}}+{x}_{{2}},{y}_{{1}}+{y}_{{2}},{z}_{{1}}+{z}_{{2}}\right)}$$.Then
$$\displaystyle{\left\lbrace\begin{array}{c} {x}_{{1}}+{x}_{{2}}={a}{t}_{{1}}+{a}{t}_{{2}}={a}{\left({t}_{{1}}+{t}_{{2}}\right)}\\{y}_{{1}}+{y}_{{2}}={b}{t}_{{1}}+{b}{t}_{{2}}={b}{\left({t}_{{1}}+{t}_{{2}}\right)}\\{z}_{{1}}+{z}_{{2}}={c}{t}_{{1}}+{c}{t}_{{2}}={c}{\left({t}_{{1}}+{t}_{{2}}\right)}\end{array}\right.}$$
Hence the components of w are of the form $$\displaystyle{x}={a}{t},{y}={b}{t},{z}={c}{t}{f}{\quad\text{or}\quad}{t}={t}_{{1}}+{t}_{{2}}$$.
Now let k be a real number. And consider $$\displaystyle{w}={k}{v}_{{1}}={\left({k}{x}_{{1}},{k}{y}_{{1}},{k}{z}_{{1}}\right)}$$.Then
$$\displaystyle{\left\lbrace\begin{array}{c} {k}{x}_{{1}}={k}{a}{t}_{{1}}={a}{\left({k}{t}_{{1}}\right)}\\{k}{y}_{{1}}={k}{b}{t}_{{1}}={b}{\left({k}{t}_{{1}}\right)}\\{k}{z}_{{1}}={k}{c}{t}_{{1}}={c}{\left({k}{t}_{{1}}\right)}\end{array}\right.}$$
Hence the components of w are of the form $$\displaystyle{x}={a}{t},{y}={b}{t},{z}={c}{t}{f}{\quad\text{or}\quad}{t}={k}{t}_{{1}}$$.
Then, by the subspace theorem, L is a subspace pf $$\displaystyle\mathbb{R}^{{3}}$$.

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