Question

find the vector and the cartesian equation of the line passing through (1,2,3) and parallel to the planes
x-y+2z=5 and 3x+y+z=6

Answer 1

that "parallel to the planes" part is a little confusing. The planes themselves have different normals so they are not even parallel to each other. And also, there are alot of lines that pass thru a point that are all parallel to a given plane.

For example, take a plane that contains the given point; how many lines can you form in that plane that go thru the point?

Answer 2

First find the equation of the plane parallel to x-y+2z=5 and that passes through (1,2,3)
Then also find equation of the plane that is parallel to 3x+y+z=6 and that passes through (1,2,3)
Then the intersection of those two plane is the required line

Answer 3

ahhh, thats what they meant; the line of intersection of the 2 planes .... good eye on that one :)

Answer 4

how..

Answer 5

@sauravshakya

Answer 6

Actually there are infinite lines that is parallel to plane and that passes thorough a point.
So, first find equation of the plane that is parallel to given plane and that passes throughh a point

Answer 7

Do similar for another given plane

Answer 8

Then intersection of those two plane will be our required line tha will be parallel to both the given planes

Answer 9

would u plse solve one

Answer 10

@sauravshakya
plse help me

Answer 11

@sauravshakya
plse help me

Answer 12

find the equation of the plane parallel to x-y+2z=5 and that passes through (1,2,3)

Answer 13

Can u find it?

Answer 14

i can't find it..

Answer 15

srry

Answer 16

Let the equation of the plane parallel to x-y+2z=5 be x-y+2z=d

Answer 17

Now, using the give point find d

Answer 18

d = 5

Answer 19

if you plug in the point values into both plane equations you will discover that the point is common to both planes; which can only happen if that point is on the line of the intersection of both planes.

if we cross the normals of the planes, we create a vector that is parallel to the line of intersection. Then just apply that vector to the given point to define the line itself