Vector-Parametric Equation of a Plane

[BeanbagxD]

Homework Statement

Find a vector-parametric equation of a plane that contains the point (4,-2,14) and is parallel to the vectors u=2i-4j+7k and v=3i+5j-2K.

Homework Equations

Vector Parametric Equation of a Plane: r=a+λu+μv
(if u and v are two non-parallel vectors in the plane)

The Attempt at a Solution

If u and v are parallel to the vector, then wouldn't one of the vectors need to be altered in order for the equation to work? If so, I found the cross product of u and v

u x v= -27i+25j+22k

But would that work? I am unsure whether it is necessary to change u and v in the first place, because I don't know if the vectors can't be parallel to each other, the unfound vector, or the plane itself. Does anyone know how this can be found?

 

[Mark44]

Homework Statement

Find a vector-parametric equation of a plane that contains the point (4,-2,14) and is parallel to the vectors u=2i-4j+7k and v=3i+5j-2K.

Homework Equations

Vector Parametric Equation of a Plane: r=a+λu+μv
(if u and v are two non-parallel vectors in the plane)

The Attempt at a Solution

If u and v are parallel to the vector, then wouldn't one of the vectors need to be altered in order for the equation to work?

Why would you think this? And what equation are you talking about? You're trying to find the equation of the plane in terms of vectors and a couple of parameters.

If so, I found the cross product of u and v

u x v= -27i+25j+22k

But would that work? I am unsure whether it is necessary to change u and v in the first place, because I don't know if the vectors can't be parallel to each other, the unfound vector, or the plane itself. Does anyone know how this can be found?

The given vectors are parallel to the plane, but obviously not parallel to each other. That can't happen in R², where two vectors parallel to a line have to be parallel to each other, but it can happen in R³.

The cross product of your two given vectors gives a vector that is normal to each of them, and hence, to the plane. If you know a point on a plane and its normal, you can find the equation of the plane, either as standard form (ax + by + cz = d) or in terms of the given point and the two vectors.

 

[LCKurtz]

Homework Statement

Find a vector-parametric equation of a plane that contains the point (4,-2,14) and is parallel to the vectors u=2i-4j+7k and v=3i+5j-2K.

Homework Equations

Vector Parametric Equation of a Plane: r=a+λu+μv
(if u and v are two non-parallel vectors in the plane)

Doesn't your problem exactly fit the relevant equation you have listed?