# Vector Equation of Plane w/ (-2,2,1) & Parallels (1,1,2) and (2,1,-1)

• kent davidge
In summary, the vector equation for a plane containing the point (-2,2,1) and whose vectors (1,1,2) and (2,1,-1) are parallel to it is -3(x+2)+5(y-2)-(z-1) = 0. An arbitrary point in the plane can be represented as P(t,u) = (t,u,-3t+5u-15) with {t,u} in the set of real numbers. This is a valid answer as it satisfies the plane equation.
kent davidge

## Homework Statement

What's the vector equation for a plane which contains the point ##(-2,2,1)## and whose vectors ##(1,1,2)## and ##(2,1,-1)## are parallel to it?

## Homework Equations

I think the relevant here is

- The plane equation.

## The Attempt at a Solution

[/B]
We can go through demanding that a vector on it can always be expressed as ##(-2,2,1) + a(1,1,2) + b(2,1,-1)## but I did in a different manner.

I first took the vector product of the two given vectors, which is ##(-3,5,-1)## and then constructed the equation of the plane, passing on the given initial point ##(-2,2,1)##. It is ##-3(x+2)+5(y-2)-(z-1) = 0##. Then I considered an arbitrary point in the plane having "coordinates" ##(a,b,c)## and I substituted these components into the equation above. Calling the "point" (vector) ##P## I ended up with ##P(t,u) = (t,u,-3t+5u-15)## with ##\{t,u \} \in \mathbb{R}##.

It seems to satisfy the plane equation as it's constructed to satisfy it. Never the less, I would like to know whether this is a valid answer or not.

You can check this by expressing a and b in terms of t and u using two of the components or vice versa. If the third component is then also the same it is the same plane.

kent davidge

## 1. What is a vector equation of a plane?

A vector equation of a plane is an equation that describes the position of all points in a three-dimensional space that lie on a flat surface. It is typically written in the form r = r0 + sa + tb, where r is a position vector, r0 is a fixed point on the plane, and a and b are direction vectors.

## 2. How do you determine if two planes are parallel?

Two planes are parallel if their normal vectors are parallel. In other words, if the direction vectors of the two planes are scalar multiples of each other, then the planes are parallel.

## 3. What is the significance of the given point (-2,2,1) in the vector equation of the plane?

The point (-2,2,1) is a fixed point on the plane, also known as the r0 in the vector equation r = r0 + sa + tb. It represents the point where the plane intersects the x, y, and z axes.

## 4. How can you find the normal vector of a plane?

The normal vector of a plane can be found by taking the cross product of the direction vectors of the plane. In this case, the normal vector would be n = a x b = (1,1,2) x (2,1,-1) = (-3,3,-1).

## 5. What is the significance of the given direction vectors (1,1,2) and (2,1,-1) in the vector equation of the plane?

The direction vectors (1,1,2) and (2,1,-1) represent two non-parallel lines on the plane. They are used to determine the orientation and size of the plane in three-dimensional space.

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