Deromediqm

Answered

2022-07-14

I have a question on the following:

The line r = i + ( 1 + 3t )j - ( 3 - 4t )k passes through ${p}_{1}=(1,1,-3)$ 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

The line r = i + ( 1 + 3t )j - ( 3 - 4t )k passes through ${p}_{1}=(1,1,-3)$ 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

Answer & Explanation

Anaya Gregory

Expert

2022-07-15Added 14 answers

$\mathbf{r}=\hat{i}+\hat{j}-3\hat{k}+t(3\hat{j}+4\hat{k}),t\in \mathbb{R}$ represents the equation of a straight line that passes through the point with position vector $\hat{i}+\hat{j}-3\hat{k}$ (take t=0). Take any two distinct points on the line, say for ${t}_{1}\ne {t}_{2}\in \mathbb{R}$

$\begin{array}{r}{\mathbf{r}}_{\mathbf{1}}=\hat{i}+\hat{j}-3\hat{k}+{t}_{1}(3\hat{j}+4\hat{k})\\ {\mathbf{r}}_{\mathbf{2}}=\hat{i}+\hat{j}-3\hat{k}+{t}_{2}(3\hat{j}+4\hat{k})\end{array}$

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 $({t}_{1}-{t}_{2})(3\hat{j}+4\hat{k})$ or put simply, the line is parallel to $3\hat{j}+4\hat{k}$

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..

$\begin{array}{r}{\mathbf{r}}_{\mathbf{1}}=\hat{i}+\hat{j}-3\hat{k}+{t}_{1}(3\hat{j}+4\hat{k})\\ {\mathbf{r}}_{\mathbf{2}}=\hat{i}+\hat{j}-3\hat{k}+{t}_{2}(3\hat{j}+4\hat{k})\end{array}$

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 $({t}_{1}-{t}_{2})(3\hat{j}+4\hat{k})$ or put simply, the line is parallel to $3\hat{j}+4\hat{k}$

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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