Intersecting Planes

[sqleung]

Hello. Could somebody please help me out on some questions about intersecting planes? This is rather urgent but any help would be appreciated

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Two vectors a and b are the normals to two planes. Both of these planes pass through (0,0,0). Vector OP = t[a xb] where t is a scalar.

1. Describe the set of all points P in relation to the two distinct planes

For this one, can somebody perhaps explain it in "simpler" terms because I'm rather confused on the wording of it.

2. From above or otherwise, find the parametric equations of the line of intersection between the two planes

5x + 2y - z = 0
3x + y + 4z = 0

Okay, first, I got the direction vectors of the two equations which are [5,2,-1] and [3,1,4].

From there, I cross-producted these two vectors to get [9,-23,-1] which is the perpendicular of the two vectors. Now vector OP is [9t,-23t,-t] where t is a scalar.

Is the above what I'm suppose to do? If so, how can I use it to find the line of intersection?

3. Show the line of intersection from before is parallel to plane 4x + y + 11= 26

Now for this one, I'll need to do the above question first and once I get it (via your assistance), I'll try it and if I cannot get it, I'll ask for help.

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Thankyou, all help is appreciated.

 

[soroban]

Hello, sqleung

\(\displaystyle \text{1. Two vectors }\vec{a}\text{ and }\vec{b}\text{ are the normals to two planes.}\)
\(\displaystyle \text{Both of these planes pass through }(0,0,0).\)
\(\displaystyle \text{Vector }\overrightarrow{OP} \,=\, t(a \times b)\text{, where }t\text{ is a scalar.}\)

\(\displaystyle \text{1. Describe the set of all points }P\text{ in relation to the two distinct planes.}\)

This was difficult to visualize and comprehend.

\(\displaystyle \text{We know that the vector: }\;\vec a\times \vec b\,\text{ is perpendicular to both normals.}\)
. . \(\displaystyle \text{It is parallel to their line of intersection.}\)

\(\displaystyle \text{Since }\overrightarrow{OP}\text{ passes through the origin and both planes pass through the origin,}\)
. . \(\displaystyle \text{then }\overrightarrow{OP}\;{\bf is}\text{ their line of intersection.}\)

\(\displaystyle \text{Therefore, the set of all points }P\text{ is the line of intersection of the two planes.}\)

\(\displaystyle \text{2. From above or otherwise, find the parametric equations of the line of intersection}\)

\(\displaystyle \text{ between the two planes: }\;\begin{array}{ccc}5x + 2y - z &=& 0 \\ 3x + y + 4z &=& 0 \end{array}\)


\(\displaystyle \text{Okay, first I got the direction vectors of the two planes which are: }\;\langle5,2,-1\rangle\text{ and }\langle3,1,4\rangle.\)

\(\displaystyle \text{From there, I cross-producted these two vectors to get: }\;\langle9,-23,-1\rangle\)
. . \(\displaystyle \text{which is perpendicular to the two normals.}\)

\(\displaystyle \text{Now: }\;\overrightarrow{OP} \;=\;[9t,-23t,-t]\text{, where }t\text{ is a scalar.}\)

Is the above what I'm suppose to do? . . . . Yes! Good work!

\(\displaystyle \text{3. Show the line of intersection from before is parallel to plane }\,4x + y + 11\:=\: 26\) .?

\(\displaystyle \text{The line of intersection has direction vector: }\;\vec v \:=\:\langle9,-23,-1\rangle\)

\(\displaystyle \text{The plane has normal vector: }\:\vec{n} \:=\:\langle4,1,0\rangle\)

\(\displaystyle \text{If the line is parallel to the plane, then: }\;\vec{n} \perp \vec{v}\)
. . \(\displaystyle \text{But: }\;\vec{n}\cdot\vec{v} \:=\:\langle 9,-23,-1\rangle\cdot\langle 4,1,0\rangle \:=\:36 - 23 + 0 \:=\:13\quad\hdots\quad\text{not perpendicular!}\)

\(\displaystyle \text{Obviously, there are typos . . .}\)

\(\displaystyle \text{I assumed that it was supposed to be: }\;4x + y + 11z \:=\:26\)
. . \(\displaystyle \text{but: }\;\langle 9,-23,-1\rangle\cdot\langle4,1,11\rangle \:=\:36 - 23 - 11 \:=\:2\quad\hdots\quad\text{not perpendicular!}\)


I give up . . . What is the question?

 

[sqleung]

Oh, nevermind, I managed to solve the problem. And yes, there was a typo:

5x + 2y - z = 0 should be:
5x + 2y + z = 0

Thankyou for your help however. It was definitely most appreciated