Proving Parallelism of Line and Plane in 2x-y+4z=81 and x-2/3=y-3/2=z-1

In summary, the problem is to show that the plane 2x - y + 4z = 81 and the line with the direction vector 3i+2j+k never intersect. However, the given direction vector and normal vector of the plane are not perpendicular, indicating that there may be a typo in the problem.
  • #1
fiziksfun
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Homework Statement





show that the plane 2x - y + 4z = 81

never intersects the line

[tex]\frac{x-2}{3}[/tex]=[tex]\frac{y-3}{2}[/tex]=z-1



Homework Equations



??


The Attempt at a Solution



I wanted to show that the line and the plane were parallel. So the unit vector for the line would be 3i + 3j + 1k
RIGHT?
Then I get confused how to show this is parallel to the plane
Planes don't have unit vectors do they ?
The vector normal to the plane i suppose is 2i + -j + 4z

So if the two were parallel a dot b would be = 0
but this doesn't work ... help
 
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  • #2
I think the unit vector of the line is 3i+2j+k. But that doesn't change anything, the direction vector and the normal vector still aren't perpendicular. That can only mean that the line and the plane must intersect. There is probably a typo in the problem.
 

1. How do you prove parallelism in a line and a plane?

To prove parallelism in a line and a plane, you need to check if the direction vector of the line is perpendicular to the normal vector of the plane. This can be done by finding the direction vector of the line and the normal vector of the plane using the coefficients of x, y, and z in the equations.

2. What is the direction vector of a line?

The direction vector of a line is a vector that shows the direction of the line. It is calculated by taking the coefficients of x, y, and z in the equation of the line. For example, in the equation x-2/3=y-3/2=z-1, the direction vector would be (1, 1, 1).

3. What is the normal vector of a plane?

The normal vector of a plane is a vector that is perpendicular to the plane. It is calculated by taking the coefficients of x, y, and z in the equation of the plane. For the plane 2x-y+4z=81, the normal vector would be (2, -1, 4).

4. How do you check if two vectors are perpendicular?

To check if two vectors are perpendicular, you need to calculate their dot product. If the dot product is equal to 0, then the vectors are perpendicular. In the case of proving parallelism of a line and a plane, the dot product of the direction vector of the line and the normal vector of the plane should be equal to 0.

5. Can parallel lines and planes intersect?

No, parallel lines and planes cannot intersect. If two lines or planes are parallel, it means they have the same direction and will never intersect. In the case of proving parallelism of a line and a plane, if the direction vector of the line is parallel to the normal vector of the plane, it means they will never intersect.

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