For what value of h is y in the plane spanned by v1 and v2?

Question

Let v1= [1,2,-1], v2=[-2,-1,1], and y=[4,-1,h]. 

For what value of h is y in the plane spanned by v1 and v2?

Answer 1

Another method, although not quite as simple as Francisco's solution, is pretty elegant and more linear algebra-y(if this is for a linear class)  is to take the determinant of the three vectors.

 

If the determinant is 0, they are linearly dependent and coplanar:

 

 

 

det (4 -1 h)

       (1 2 -1)

       (-2 -1 1)

 

=   4 ( 2  - 1) + 1(1 - 2) +  h(-1 + 4) = 0

 

4 - 1 + 3h = 0

 

h = - 1

 

Same solution as you can see.

Answer 2

For the plane spanned by v₁ and v₂, y is a linear combination of v₁ and v₂ if it is in the plane.

 

y = av₁ + bv₂ where a and b are constants.

 

[4,-1,h] = a[1,2,-1] + b[-2,-1,1]

 

[4,-1,h] = [a - 2b,2a - b,-a + b] by the rules of vector addition and multiplication.  Vectors are equal if their components are equal:

 

4 = a - 2b

 

-1 = 2a - b

 

h = -a + b

 

 

We can solve for a and b from the first two equations to find h:

 

a = -2

 

b = -3

 

h = -1

 

 

We see that -2[1,2,-1] - 3[-2,-1,1] = [4,-1,-1].