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A linear combination of vectors involves combining two or more vectors by multiplying them by scalars and then adding the results together. This is a key concept when determining if a vector lies within a specific plane formed by other vectors.
A vector can be expressed as a linear combination of two vectors \(\vec{a}\) and \(\vec{b}\) if it can be written in the form \(p\vec{a} + q\vec{b}\), where \(p\) and \(q\) are scalars. In this problem, we want to express vector \(\vec{c} = x \hat{i} + (x-2) \hat{j} - \hat{k}\) as such a combination.

• By finding weights \(p\) and \(q\) that satisfy the equation \(p\vec{a} + q\vec{b} = \vec{c}\), you determine the required combination.
• This relationship proves if the vector truly lies in the same plane formed by the other vectors.
• Linear combinations are fundamental in understanding vector spaces and planes within vector mathematics.

Understanding this concept helps solve problems involving vector equations and is a foundation for further study of vector algebra and linear algebra.

The vector plane in a three-dimensional space is essentially a two-dimensional flat surface that can be defined by two non-parallel vectors. In this regard, \(\vec{a}\) and \(\vec{b}\) from our problem define such a plane, and we are interested if \(\vec{c}\) falls on this plane.
To check if a vector lies in a particular plane, verify that it can be written in terms of the plane-defining vectors. In our example, \(\vec{a} = \hat{i} + \hat{j} + \hat{k}\) and \(\vec{b} = \hat{i} - \hat{j} + 2\hat{k}\) create the plane on which we want to see if \(\vec{c}\) resides.

• If a vector \(\vec{c}\) is in the plane, it can be described as a linear combination of the vectors that form the plane, meaning \(\vec{c}\) is co-planar with \(\vec{a}\) and \(\vec{b}\).
• This conversion into a linear combination forms the core method to verify the position of a vector in the vector plane.

Understanding the concept helps evaluate vector relationships and solve geometry problems in vector spaces including determining dependence among vectors.

A system of equations is a set of equations with the same variables. Solving such a system involves finding the values of the variables that satisfy all equations concurrently.
In this problem, the coefficients of the linear combination of vectors \(\vec{a}\) and \(\vec{b}\) are matched to \(\vec{c}\). Hence, this yields a system of equations:

• Equation 1: \(p + q = x\)
• Equation 2: \(p - q = x - 2\)
• Equation 3: \(p + 2q = -1\)

Solving these equations helps to find values for \(p, q,\) and \(x\) such that all three conditions are satisfied,
which ensures \(\vec{c}\) is a true linear combination of \(\vec{a}\) and \(\vec{b}\).
Using simple algebra:

• Adding equations can eliminate variables to simplify finding another.
• Substitution helps to solve for the unknowns by providing a possible derived expression that relates them.

This process aids students in mastering problem-solving strategies involving vectors and is essential in exploring more advanced topics such as linear transformations.