Question

Let \(A\) and \(B\) be points other than the origin, and let \(\mathbf{a}\) and \(\mathbf{b}\) be their vectors. If \(\mathbf{a}\) and \(\mathbf{b}\) are not parallel, show that the plane through \(A, B\), and the origin is given by $$ \left\\{P(x, y, z) \mid\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=s \mathbf{a}+t \mathbf{b} \text { for some } s \text { and } t\right\\} $$

Short answer

Any point on the plane is a linear combination of \( \mathbf{a} \) and \( \mathbf{b} \).

Step-by-step solution

Express Planar Points as Vectors

In vector space, a plane can be described using three non-collinear points. Here, we are given the origin (denote as \( \mathbf{0} \)), point \( A \) with vector \( \mathbf{a} \), and point \( B \) with vector \( \mathbf{b} \), which are not parallel, thus ensuring they form a plane.

Use Linear Combination for Plane Description

Any point \( P(x, y, z) \) on the plane passing through the origin, \( A \), and \( B \) can be written as a linear combination of the vectors \( \mathbf{a} \) and \( \mathbf{b} \). This means \( \mathbf{p} = s\mathbf{a} + t\mathbf{b} \) for some scalars \( s \) and \( t \).

Substitute Vector Expression for Points on Plane

Substitute the expression from Step 2 into point \( P(x, y, z) \): \( \mathbf{p} = \begin{bmatrix} x \ y \ z \end{bmatrix} = s \begin{bmatrix} a_1 \ a_2 \ a_3 \end{bmatrix} + t \begin{bmatrix} b_1 \ b_2 \ b_3 \end{bmatrix} \).

Validate the Plane Equation

By substituting the vectors of \( \mathbf{a} \) and \( \mathbf{b} \) into the expression for \( \mathbf{p} \), you achieve the general linear combination representing all points \( P(x, y, z) \) on the plane. Hence, every point on this plane satisfies \( \mathbf{p} = s\mathbf{a} + t\mathbf{b} \), confirming the statement of the problem.

Key concepts

Plane Geometry

Plane geometry involves shapes and figures in a flat, two-dimensional space. When dealing with vector spaces, plane geometry helps us understand how points and vectors lay in a plane.
In this case, when we mention a 'plane through the origin, point A, and point B', it means these three points create a surface in space. The origin acts as a central point, while points A and B (with their associated vectors \(\mathbf{a}\) and \(\mathbf{b}\)) define the directions within this plane.
Understanding this helps visualize geometric figures not as isolated entities but as part of the great landscape of geometry. Analyzing how these figures interact enables us to explore relationships between points and vectors in a spatial context.

Linear Combination

A linear combination in mathematics is a way of combining two or more vectors using scalars. Scalars are simple numerical values that multiply the vectors without changing their direction.
For a point \( P(x, y, z) \) on the plane, we can say that it is a linear combination of vectors \( \mathbf{a} \) and \( \mathbf{b} \).

• This means that we express \( P \) as \( s\mathbf{a} + t\mathbf{b} \), where \( s \) and \( t \) are scalars.
• This forms all possible points on the plane by adjusting \( s \) and \( t \).

The significance of linear combinations is that they allow us to describe an infinite number of points on a plane precisely and systematically.

Vector Representation

Vectors are mathematical objects that have both magnitude and direction. They are often represented as arrows pointing from one location to another, and in algebraic terms, as ordered pairs or triples.
In this example, vectors \( \mathbf{a} \) and \( \mathbf{b} \) represent points A and B in space. These vectors determine the orientation and extent of the plane.

• Vectors provide a way to translate geometric interpretations into algebraic expressions.
• The notation \( \begin{bmatrix} x \ y \ z \end{bmatrix} = s \mathbf{a} + t \mathbf{b} \) shows how vector operations help locate every point \( P \) on the plane.

Using vectors compacts complex geometric ideas into manageable mathematical expressions.

Non-Collinear Points

Non-collinear points are points that do not all lie on a single straight line. Their importance in plane geometry is vital because three non-collinear points define a unique plane.
In this problem, points at the origin, A, and B are non-collinear, meaning they can span an area in space, forming a plane.

• This ensures the vectors \( \mathbf{a} \) and \( \mathbf{b} \) are not parallel.
• In linear algebra terms, this independence is crucial for forming comprehensive vector spaces.

Grasping this concept clarifies the spatial relationships among points and guarantees that the vectors explain a true and complete plane, rather than a line.