Calc 3 Need help to calculate distance in 3-space

Hey need some help with some questions

1.)Consider the points P = (3,-1,2) , Q=(4,1,-5) , and R=(-1,3,3).

a. Compute the distance from the Origin O=(0,0,0) to the line PQ.
b. Compute the distance from the origin to the plane PQR.
c. Compute the distance between the lines PQ and OR.

I "think" the formula is

$\displaystyle |U x V| / |v|$

so far a.)

PQ = < 1, 2 , -7 >

So if my formula is correct I believe PQ would be V. What would be U? <0,0,0> ?

b.)
PQR = <2,-1, -10> ?
for c.)

PQ = <1,2,-7>
OR = <-1,3,3>




Any help in the right direction or if you could list the process would be of great help.

Hey Jay23456.

Just before I go further, are all these distances perpendicular distances?

It does not state but I would assume so, because I think then we did it in class we drew a right angle.

BTW here are my answers

a.) sqrt(587)/ sqrt(54)

b. 87 / sqrt(1773)

c. sqrt(105)

If you are using formulas to do this without knowing what they mean then this will be detrimental to you in the future.

In the first example, you are finding the minimum distance from the origin to a given line.

So what you can do is find a plane where the line from the origin to the original line is perpendicular and this is what this formula actually does. You then have an equation that describes the distance from a point to a plane and you use this to get your actual distance.

When you plug a point into the plane equation n . (r - r0), if the normal is a unit normal then that will return the distance to the actual plane and if it the point is on the plane it returns 0 but if not it returns the perpendicular distance from a point ot the plane.

For 2 its the same sort of thing, but for 3 you need to find a vector that is perpendicular to both lines (which involves the cross product as well) but then you need to the point where the normal distance is minimized.

Just out of curiosity, how have you been taught how to do all this? Do they just give you formulas and you just use them without you knowing how it all works?

He demonstrated by constructing a right triangle from the line segments and used the magnitude of one of the line segments and multiplied it by the components of a side. It was very fast and I tried to look in my book but I could only find few examples.