Hello so 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
###thanks in advance###
$\mathbf r=\hat i+\hat j-3\hat k+t(3\hat j+4\hat k),t\in\Bbb R$ 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 $t_1\ne t_2\in\Bbb R$,$$\begin{align*}\mathbf{r_1}=\hat i+\hat j-3\hat k+t_1(3\hat j+4\hat k)\\\mathbf{r_2}=\hat i+\hat j-3\hat k+t_2(3\hat j+4\hat k)\end{align*}$$Then $\bf r_1-r_2$ 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..