Calc 3 Need help with finding distances in 3-space

[John45]

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.

 

[tkhunny]

a) No. That's a point, not a vector. You need the point on PQ, call it T, where OT is perpendicular to PQ. Can you write a generalized expression for ALL points on PQ (possiby extended)?

 

[pka]

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 >

As pointed out in reply #2 that formula is incorrect.
Let \(\displaystyle D=<1,2,-7>\). That is the direction vector of the line.
Thinking of a position vector \(\displaystyle \overrightarrow P = \left\langle {3, - 1,2} \right\rangle \) the distance the origin is from the line equals
\(\displaystyle \left\| {\overrightarrow P - \frac{{\overrightarrow P \cdot D}}{{D \cdot D}}D} \right\|\).