Express the following plane in vector form: P 1 ⊆ R 3 with equation 4...
Express the following plane in vector form:
The answer is . I don't understand how they got (0,1,0) for the direction vector.
Answer & Explanation
The cartesian equation is z=4x
The parametrization is not unique. Anyway You need two linearly independent points on the plane.
One point can be A=(1,0,4). The second one cannot be (2,0,8) or other points you can get from A multiplying by a constant. Otherwise you get a line not a plane.
The second point cannot obviously be (0,0,0)
So we look for a point where the only nonzero coordinate is y, like (0,1,0) or (0,10,0)
Finally the parametric equation of the plane is
Note that would be right.
Hope this helps
Let be your plane. We notice that the point and the orthogonal vector to the plane is
The vector space has dimension 2 and the vectors that span this space are orthogonal to the vector
We can take for example the vectors and , infact we have
Finally the equation of the plane is