the distance from the point -i + 2j + 6k to the straight line through the point (2,3,-4) and parallel to the vector 6 i+ 3j - 4k.

- yrelhan4

- chestercat

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- anonymous

Seems like a work for the dot product to me.

- yrelhan4

so i converted all of it into cartesian form. wrote the line as (x-2)/6=(y-3)/3=(z+4)/-4
then i put all of it =r. found out random points. then found out direction ratios of the perpendicular. through condition of perpendicularity found r. and found those random points. and the found the distance.
the answer i am getting is wrong.

- anonymous

|dw:1360345934670:dw|

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## More answers

- anonymous

careful, a line in 3D space shouldn't be written in cartesian form,

- anonymous

\[\Large \vec{v} \cdot \vec{SP}= 0 \]

- anonymous

\[\Large 0S \in g \]

- anonymous

|dw:1360346169283:dw|

- yrelhan4

well it can be written in cartesian form with direction ratios.

- anonymous

well if you want to do it like that, then unfortunately I can't help you. I haven't seen such a thing, what I would agree with, is saying that you can always set up an equation of a plane that includes the line and from there figure out the distance.
But as soon as I see something of the form x+y+z=c, I think of a plane, not of a line.

- TuringTest

I am confused for many reasons:
"the distance from the point -i + 2j + 6k..." but -i +2j + 6k is not a point.
and the distance to which point on the other line? the minimum distance I presume?

- yrelhan4

hmm. i just want to confirm if i am interpreting the question right. its saying the line is parallel to the vector 6 i+ 3j - 4k and not that it wants the distance parallel to this vector?

- anonymous

Mathematical definition is always considering it's normal, orthogonal point. That is a distance. So if you have a point given, as I tried to sketch it in the above picture, then it's distance to the line is the one where you start from the line, draw a right angle through that point and figure out the distance.

- yrelhan4

@TuringTest well its the position vector as they call it. its basically a point.

- TuringTest

ok, I think I understand the question better now, thanks

- yrelhan4

@Spacelimbus well i still have the doubt. does it ask for the distance parallel to that vector? or the line is parallel to that vector? 6 i+ 3j - 4k.

- TuringTest

the line is parallel to the vector in my interpretation

- sirm3d

|dw:1360346673680:dw|

- anonymous

To set up an equation in the 3D-Space, you need a direction vector, an a point, that's how I interpret it, because that satisfies the equation:
\[\Large r_x=0P + \lambda \vec{v} \]

- anonymous

I guess @sirm3d and I agree then, 6i+4j-4k is a direction vector of the parametric line

- sirm3d

as @Spacelimbus said, it's the dot product at work here.

- TuringTest

I agree as well, for what that's worth :p

- yrelhan4

hmm. and whats wrong with the method i'm using?

- anonymous

|dw:1360346938724:dw|

- anonymous

stealing @sirm3d superior model

- sirm3d

|dw:1360347140886:dw|
\[\cos \theta = \frac{\vec{PR} \cdot \vec{SR}}{|\vec{PR}||\vec{SR}|}\\|\vec{PS}|=|\vec{PR}|\sin\theta\]

- yrelhan4

i think i can do it now. thank you guys.

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