Deromediqm

2022-07-14

I have a question on the following:
The line r = i + ( 1 + 3t )j - ( 3 - 4t )k passes through ${p}_{1}=\left(1,1,-3\right)$ and is parallel to v = 3j + 4k.
But why can we say that this is parallel? And how can I apply this for different problems.
If someon could help me, it would be very much appreciated

Anaya Gregory

Expert

$\mathbf{r}=\stackrel{^}{i}+\stackrel{^}{j}-3\stackrel{^}{k}+t\left(3\stackrel{^}{j}+4\stackrel{^}{k}\right),t\in \mathbb{R}$ represents the equation of a straight line that passes through the point with position vector $\stackrel{^}{i}+\stackrel{^}{j}-3\stackrel{^}{k}$ (take t=0). Take any two distinct points on the line, say for ${t}_{1}\ne {t}_{2}\in \mathbb{R}$
$\begin{array}{r}{\mathbf{r}}_{\mathbf{1}}=\stackrel{^}{i}+\stackrel{^}{j}-3\stackrel{^}{k}+{t}_{1}\left(3\stackrel{^}{j}+4\stackrel{^}{k}\right)\\ {\mathbf{r}}_{\mathbf{2}}=\stackrel{^}{i}+\stackrel{^}{j}-3\stackrel{^}{k}+{t}_{2}\left(3\stackrel{^}{j}+4\stackrel{^}{k}\right)\end{array}$
Then ${\mathbf{r}}_{\mathbf{1}}\mathbf{-}{\mathbf{r}}_{\mathbf{2}}$ is a vector that is parallel to the line, i.e. the line is parallel to $\left({t}_{1}-{t}_{2}\right)\left(3\stackrel{^}{j}+4\stackrel{^}{k}\right)$ or put simply, the line is parallel to $3\stackrel{^}{j}+4\stackrel{^}{k}$
The information about a vector that is parallel to a line is as important as the information about the normal to a plane. It comes in handy in almost all applications involving lines, such as finding the distance of a point from the line, finding the image of a point with respect to a line, finding the angle between a line and a plane, etc..

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