find the line that passes through the point (2,5,3) and is perpendicular to the plane 2x-3y+4z+7 = 0

Question

turingtest · Answer

this problem is almost identical to the last, so I'm gonna make you do a lot more on your own this time ;)
1) what is $\vec n$ of the given plane?

anonymous · Answer

(2,-3,4) ?

turingtest · Answer

yes, nice
I just realized that this problem is a bit different though, because it asks for the equation of a normal line, not for the plane
similar business though; we still needed the normal vector
now we need a line that is parallel to that vector

I don't suppose that you know that the equation of a line through a point $P_0$ is given by$$\vec r(t)=\vec r_0+t\vec v$$where $\vec r_0$ is the position vector of the given point, and $\vec v$ is a vector parallel to the line, but that is supposed to be know to solve this problem.

I will try to help you understand it anyway though...

turingtest · Answer

here is a link on the ideas I will express in my explanation http://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfLines.aspx please open it and look at the picture of the red line

turingtest · Answer

in this case to use our formula we need to not that the parallel vector in the line formula $\vec v=\vec n$ because we want a line perpendicular to the plane, which implies it must be parallel to $\vec n$
because of that we can rewrite this as$$\vec r=\vec r_0+t\vec n$$now you already know $\vec n$, so now tell me
what is $\vec r_0$ (the position vector that points to the given point)

anonymous · Answer

(2,5,3)?

turingtest · Answer

yep, so now you can just plug int the stuff we know
why don't you give that a try?

anonymous · Answer

= (2,5,3)+(2,-3,4) ?

turingtest · Answer

don't forget the t, which we treat with scalar multiplication (scalar multiplication is this:$c\vec v=t=$)$$\vec r(t)=\vec r_0+t\vec n$$$$\vec r(t)=<2,5,3>+t<2,-3,4>$$$$\vec r(t)=<2+2t,5-3t,3+4t>$$

anonymous · Answer

thanks

anonymous · Answer

getting it slowly

turingtest · Answer

by looking at the individual components we can also write this as$$x=2+2t$$$$y=5-3t$$$$z=3+4t$$there are a number of ways to write the equation of a line

anonymous · Answer

thanks.. next question?

turingtest · Answer

give me a few minuts, I am eating breakfast and my cat just broke something :(

anonymous · Answer

haha ok no prob, im goin to take a small break, i will post a new thread for numbr 3, plz reply wen u available..