distance between point and line

[eckimz]

distance between P(3, 5, 7) and r= <1+5t, 2-2t, 2+3t>



Professor put this on the test, so I suposse I need to use that?!

So I get that [MATH]v=<5, -2, 3>[/MATH];

[MATH]PQ[/MATH] is [MATH]P(3,5,7)*Q(1, 2, 2)[/MATH]
So I did the determinant and got: [MATH]-19i-19j+19k[/MATH]
So, [MATH]V*PQ[/MATH] do I need to do the determinant or should I just multiply [MATH]<5, -2, 3>*<-19i-19j+19k>[/MATH]?

By doing that, if its right, I got ,[MATH]|(-95i+38j+57k/(sqr38)|[/MATH] where [MATH]|v|= sqr [(5)^2+(-2)^2+(3)^2]= sqr 38[/MATH]
With that, I got [MATH]sqr 13718/|sqr 38|=19[/MATH]
I have no idea, if that was righ, although I got 19.

I tried to find a video or something showing how to do with parametrics, but I can only find with with general or using two points, idk, I have a test in 11h from now, if someone can give me the steps.

I also did another way. Might be right, I'll share.

distance between P(3, 5, 7) and r= <1+5t, 2-2t, 2+3t>
[MATH]A(1, 2, 2) V(5, -2, 3) AP=(3-1, 5-2, 7-3) AP=(2, 3, 5)[/MATH]
[MATH]V.AP=[/MATH] did the determinant and got [MATH]-19i-19j+19k[/MATH]
[MATH]d= sqr [(-19)^2+(-19)^2+(19)^2]/[sqr of (5)^2+(-2)^2+(3)^2][/MATH][MATH]sqr of 1083 / sqr of 38 = 5,33[/MATH]
Is any of that right or on the way, I really need some step by step, don't wanna miss this on my test tomorrow, seems not to be difficult

 

[eckimz]

NEVER MIND, FOUND IT!

 

[pka]

distance between P(3, 5, 7) and r= <1+5t, 2-2t, 2+3t>

View attachment 12935

Professor put this on the test, so I suposse I need to use that?!

I am very sure (I think) that surely your professor gave you a correct formula.
Suppose that \(\displaystyle \ell(t)=P+t(\vec{D})\) is a line and the point \(\displaystyle Q\notin \ell\)
The the distance of \(\displaystyle Q\) to \(\displaystyle \ell\) is \(\displaystyle d(Q,\ell)=\frac{\left|\overrightarrow {PQ} \cdot\vec{D}\right|}{\|\vec{D}\|}\)
I suspect that you read the dot-product as a cross-product. It is not a cross-product.

 

[eckimz]

Yes, thank you! you've been helping me a lot, I do usually get lost on basic/simple stuff! Pretty amazing your knowledge!
Took me some hours to understand this, but now Im positive.