\(\newcommand{\R}{\mathbb{R}} \newcommand{\va}{\vec{a}} \newcommand{\vb}{\vec{b}} \newcommand{\vc}{\vec{c}} \newcommand{\vC}{\vec{C}} \newcommand{\vd}{\vec{d}} \newcommand{\ve}{\vec{e}} \newcommand{\cursedihat}{\hat{\dot{i}}} \newcommand{\vi}{\hat{\imath}} \newcommand{\vj}{\hat{\jmath}} \newcommand{\vk}{\hat{k}} \newcommand{\vn}{\vec{n}} \newcommand{\vm}{\vec{m}} \newcommand{\vr}{\vec{r}} \newcommand{\vs}{\vec{s}} \newcommand{\vu}{\vec{u}} \newcommand{\vv}{\vec{v}} \newcommand{\vw}{\vec{w}} \newcommand{\vx}{\vec{x}} \newcommand{\vy}{\vec{y}} \newcommand{\vz}{\vec{z}} \newcommand{\vzero}{\vec{0}} \newcommand{\vF}{\vec{F}} \newcommand{\vG}{\vec{G}} \newcommand{\vH}{\vec{H}} \newcommand{\vR}{\vec{R}} \newcommand{\vT}{\vec{T}} \newcommand{\vN}{\vec{N}} \newcommand{\vL}{\vec{L}} \newcommand{\vB}{\vec{B}} \newcommand{\vS}{\vec{S}} \newcommand{\proj}{\text{proj}} \newcommand{\comp}{\text{comp}} \newcommand{\nin}{} \newcommand{\vecmag}[1]{\left\lVert #1\right\rVert} \newcommand{\grad}{\nabla} \newcommand\restrict[1]{\raise-.5ex\hbox{$\Big|$}_{#1}} \DeclareMathOperator{\curl}{curl} \DeclareMathOperator{\divg}{div} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Print preview

Highlight workspace

Activity 9.5.2.

Let $P_1 = (1,2,-1)$ and $P_2 = (-2,1,-2)$ and let $\mathcal{L}$ be the line in $\R^3$ through $P_1$ and $P_2\text{.}$ Note that Figure 9.5.9 shows a similar example of a line in 3D defined by two points.

🔗

(a)

Find a direction vector for the line $\mathcal{L}\text{.}$

🔗

(b)

Find a vector equation of $\mathcal{L}$ in the form $\vr(t) = \vr_0 + t\vv\text{.}$

🔗

(c)

Consider the vector equation $\vs(t) = \langle -5, 0, -3 \rangle + t \langle 6, 2, 2 \rangle.$ What is the direction of the line given by $\vs(t)\text{?}$ Is this new line parallel to line $\mathcal{L}\text{?}$

🔗

(d)

Do $\vr(t)$ and $\vs(t)$ represent the same line, $\mathcal{L}\text{?}$ Write a couple of sentences to justify why you think $\vr(t)$ and $\vs(t)$ do or do not describe the same set of points.

🔗

Prev Top Next

Print preview



Activity 9.5.2.

(a)

(b)

(c)

(d)