Planes, lines, points

**ramzouzy** · March 2nd 2008, 01:08 PM

One question, 3 parts

A. Find an equation for the plane consisting of all points (x,y,z) that are equidistant from the points (-4,2,1) and (2,-4,3).

B. Find the points A on the line with symmetric equations x=y=z and B on that with equations x+1=y/2=z/3 such that |AB| is the shortest distance between the two lines. Hence compute the shortest distance.

C. Let L be the line of intersection of the two planes with equations x+2y-z=2 and 2x-y+4z=5. Find the point A on this line such that the distance from (0,0,0) to A is the shortest from 0 to L.

needdd urgent helppp cant find solutions to these types of problems anywhere!

**Soroban** · March 2nd 2008, 02:03 PM

Hello, ramzouzy!

Here's the first one . . .

A. Find an equation for the plane consisting of all points $P(x,y,z)$
that are equidistant from the points $A(-4,2,1)\text{ and }B(2,-4,3).$

We want $\overline{PA} \:= \:\overline{PB}.$

We have: . $\begin{array}{ccc}\overline{PA} &=& \sqrt{(x+4)^2 + (y-2)^2 + (z-1)^2} \\\overline{PB} &=& \sqrt{(x-2)^2 + (y+4)^2 + (z-3)^2} \end{array}$

Hence: . $(x+4)^2 + (y-2)^2 + (z-1)^2 \;=\;(x-2)^2 + (y+4)^2 + (z-3)^2$

. . $x^2+8x+16 + y^2-4y+4+z^2-2z+1 \;=\;x^2-4x + 4 + y^2 + 8y + 16 + z^2-6z+9$

. . $12x - 12y + 4z \:=\:8\quad\Rightarrow\quad\boxed{3x - 3y + 2z \;=\;2}$

**ramzouzy** · March 2nd 2008, 03:23 PM

anyone else for the other two parts plzzz??

**Plato** · March 2nd 2008, 04:01 PM

I will comment on the particular problem, #2. The problem is almost identical to one found in Stewarts’ Caculus. The difference is Steward asks students to find the distance between the two skew lines. That is an easy operation. But to find the particular points is rather difficult (maybe just long).
Given two skew lines $l_1 :r_1 + tD_1 \,\& \,l_2 :r_2 + tD_2 \,$ the distance between the two lines is given by $\frac{{\left| {\left( {D_1 \times D_2 } \right) \cdot \left( {r_1 - r_2 } \right)} \right|}}{{\left\| {D_1 \times D_2 } \right\|}}$ .

But finding A & B is a different matter.

**ramzouzy** · March 2nd 2008, 04:11 PM

i have both stewart single and multi variable book.. however i couldnt find the topic in the book... at some examples of the same questions or anything close to it...

and regarding that formula, is that a generic formula for skew lines or is it derived and put into place for this situation only?

**Plato** · March 2nd 2008, 05:19 PM

Originally Posted by ramzouzy

i have both stewart single and multi variable book.. however i couldnt find the topic in the book... at some examples of the same questions or anything close to it...
and regarding that formula, is that a generic formula for skew lines or is it derived and put into place for this situation only?

In Stewart 5e it is on page 867.
The distance formula I gave you is what it is: the distance between two skew lines.

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