# Show the line of intersection of both planes by finding the coordinates of a point

## Homework Statement

The planes ∏1 and ∏2 have equations 3x - y - z = 2 and x + 5y + z = 14 respectively. Show that the point (3,1,6) lies on both planes.

The Question:
By finding the coordinates of another point lying in both planes, or otherwise, show that the line of intersection of ∏1 and ∏2 has equation r = 3i + j + 6k + t ( i -j + 4k ).

## Homework Equations

Show point (3,1,6) lies in both planes:
Substitute point (3,1,6) into the r of both plane equations.

## The Attempt at a Solution

The first part:

Let point (3,1,6) be a

Substitute a into r of both plane equations.

The dot product of a and r = ''d" of the plane equations.
LHS=RHS [Shown]

How do I attempt the second part of the question? I do not quite understand the second part of the question.
Thanks!

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LCKurtz
Homework Helper
Gold Member

## Homework Statement

The planes ∏1 and ∏2 have equations 3x - y - z = 2 and x + 5y + z = 14 respectively. Show that the point (3,1,6) lies on both planes.

The Question:
By finding the coordinates of another point lying in both planes, or otherwise, show that the line of intersection of ∏1 and ∏2 has equation r = 3i + j + 6k + t ( i -j + 4k ).

## Homework Equations

Show point (3,1,6) lies in both planes:
Substitute point (3,1,6) into the r of both plane equations.

## The Attempt at a Solution

The first part:

Let point (3,1,6) be a

Substitute a into r of both plane equations.

The dot product of a and r = ''d" of the plane equations.
LHS=RHS [Shown]

How do I attempt the second part of the question? I do not quite understand the second part of the question.
Thanks!
You just need to find x,y, and z that work in both equations. You have 2 equations in 3 unknowns, so it should be easy with an extra unknown. Try something like letting y = 0 and see if you can find an x and z that work.

HallsofIvy
Homework Helper

## Homework Statement

The planes ∏1 and ∏2 have equations 3x - y - z = 2 and x + 5y + z = 14 respectively. Show that the point (3,1,6) lies on both planes.

The Question:
By finding the coordinates of another point lying in both planes, or otherwise, show that the line of intersection of ∏1 and ∏2 has equation r = 3i + j + 6k + t ( i -j + 4k ).

## Homework Equations

Show point (3,1,6) lies in both planes:
Substitute point (3,1,6) into the r of both plane equations.

## The Attempt at a Solution

The first part:

Let point (3,1,6) be a

Substitute a into r of both plane equations.

The dot product of a and r = ''d" of the plane equations.
LHS=RHS [Shown]

How do I attempt the second part of the question? I do not quite understand the second part of the question.