19-26 Find a parametric representation for the surface. The plane through the origin that c

step 1 For this problem, we are asked to find a parametric representation for the plane through the ori

19-26 Find a parametric representation for the surface. ...

$19-26$ Find a parametric representation for the surface.
\begin{equation}
\begin{array}{l}{\text { The plane through the origin that contains the vectors } \mathbf{i}-\mathbf{j}} \\ {\text { and } \mathbf{j}-\mathbf{k}}\end{array}
\end{equation}

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Key Concepts

Basis and Linear Independence

A pair of vectors forms a basis for a plane if they are linearly independent, meaning neither vector can be written as a multiple of the other. This guarantees that the linear combination using these vectors covers the entire plane. Since the plane in the problem is specified as containing the vectors, and these vectors are not multiples of each other, they form a valid basis for the plane.

Parametric Representation

A parametric representation of a plane involves expressing any point on the plane as a combination of a point on the plane and two independent direction vectors. In general, if a plane passes through a point r? and is spanned by two non-collinear vectors a and b, then every point r on the plane can be written as r = r? + s*a + t*b, where s and t are real parameters.

Span and Linear Combination

The concept of span refers to the set of all possible vectors that can be formed by taking linear combinations of a given set of vectors. In this context, the two given vectors, when combined with scalar multipliers, describe every vector in the plane, assuming these vectors are linearly independent.

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