Problem 36 - Essential Calculus Early Transcendentals

Question

Suppose that a and b are nonzero vectors that are not paral- lel and \(c\) is any vector in the plane determined by a and b. Give a geometric argument to show that \(\mathbf{c}\) can be written as \(\mathbf{c}=s \mathbf{a}+t \mathbf{b}\) for suitable scalars \(s\) and \(t .\) Then give an argument using components.

Step-by-Step Solution

Answer

Vector \( \mathbf{c} \) can be written as \( \mathbf{c} = s \mathbf{a} + t \mathbf{b} \) because \( \mathbf{a} \) and \( \mathbf{b} \) form a basis for the plane.

Step 1: Understanding the Plane

Given two nonzero and non-parallel vectors \( \mathbf{a} \) and \( \mathbf{b} \), they determine a plane in three-dimensional space. Geometrically, any vector \( \mathbf{c} \) lying within this plane can be described using a combination of vectors \( \mathbf{a} \) and \( \mathbf{b} \). This is because we can use these two vectors as a basis for the plane, similar to how any point in 2D space can be described with an \(x,y\) coordinate.

Step 2: Geometric Argument

Imagine placing the tail of vector \( \mathbf{c} \) at the origin. Since \( \mathbf{c} \) lies in the plane formed by \( \mathbf{a} \) and \( \mathbf{b} \), you can move along vector \( \mathbf{a} \) by a certain amount, and then along vector \( \mathbf{b} \) by another certain amount, to reach \( \mathbf{c} \). This means there are some scalars \( s \) and \( t \) such that \( \mathbf{c} = s \mathbf{a} + t \mathbf{b} \). This represents the idea that any vector within the plane can be reached or expressed as a combination of these two vectors.

Step 3: Argument Using Components

Assign coordinates to vectors \( \mathbf{a} = (a_1, a_2, a_3) \), \( \mathbf{b} = (b_1, b_2, b_3) \), and \( \mathbf{c} = (c_1, c_2, c_3) \). Then the equation \( \mathbf{c} = s \mathbf{a} + t \mathbf{b} \) can be expanded into component form as: \[ (c_1, c_2, c_3) = s(a_1, a_2, a_3) + t(b_1, b_2, b_3) \]. This gives us three equations: 1. \( c_1 = sa_1 + tb_1 \) 2. \( c_2 = sa_2 + tb_2 \) 3. \( c_3 = sa_3 + tb_3 \)These show that the components of \( \mathbf{c} \) can be achieved by appropriate values of \( s \) and \( t \), proving that \( \mathbf{c} \) is a linear combination of \( \mathbf{a} \) and \( \mathbf{b} \).

Step 4: Linear Independence Check

Since \( \mathbf{a} \) and \( \mathbf{b} \) are not parallel, they are linearly independent. This independence means no vector in this plane can be a multiple of another, thereby confirming that just the right choice of \( s \) and \( t \) can express any vector \( \mathbf{c} \) in the plane as a unique linear combination of \( \mathbf{a} \) and \( \mathbf{b} \).

Key Concepts

Linear Combination

In simple terms, a linear combination involves scaling and adding vectors to create a third vector. Imagine vectors as arrows in space.
When you take two or more of these arrows and combine them by scaling (making them longer or shorter) and then adding them together, you are forming a linear combination.
This concept is crucial in vector addition as it helps determine a vector's position within a space defined by other vectors.

• For any given vector \( \mathbf{c} \) that lies within the same plane as vectors \( \mathbf{a} \) and \( \mathbf{b} \), you can express \( \mathbf{c} \) as \( \mathbf{c} = s \mathbf{a} + t \mathbf{b} \), where \( s \) and \( t \) are scalars.
• The scalars \( s \) and \( t \) stretch or shrink the vectors \( \mathbf{a} \) and \( \mathbf{b} \) respectively, and determine how much of each vector is used in forming \( \mathbf{c} \).

Linear combinations are not limited to just two vectors; they can involve any number of vectors. This flexibility makes them powerful tools in geometry and linear algebra, aiding in describing spaces of various dimensions.

Plane Geometry

In the context of vectors, plane geometry refers to the study of flat, two-dimensional surfaces. When you have vectors \( \mathbf{a} \) and \( \mathbf{b} \) that are not parallel, they determine a plane. Think of this plane like a sheet of paper; any vector on this "sheet" can be described using \( \mathbf{a} \) and \( \mathbf{b} \).
This ability to describe any point on the plane using vectors simplifies understanding their geometry.

• Each vector in this plane can be imagined as a journey starting at the origin, moving in one direction along \( \mathbf{a} \) and then in another direction along \( \mathbf{b} \).
• This motion can be precisely represented as a linear combination of \( \mathbf{a} \) and \( \mathbf{b} \), with scalars determining the travel amount along each vector.

Using plane geometry, complex paths and shapes can easily be broken down into simpler components, enabling easier analysis and solving of geometric problems.

Linear Independence

Linear independence is a fundamental property in vector analysis. When vectors are linearly independent, no vector in the group can be written as a combination of the others.
This unique trait ensures that each vector adds a new direction or dimension to the space, important in forming coordinate systems in geometry.

• For example, vectors \( \mathbf{a} \) and \( \mathbf{b} \) are not parallel, indicating they are linearly independent. Because they don't lie on the same line, neither can be a scalar multiple of the other.
• This independence allows \( \mathbf{a} \) and \( \mathbf{b} \) to create a plane, where any vector like \( \mathbf{c} \) can be uniquely expressed as \( \mathbf{c} = s\mathbf{a} + t\mathbf{b} \).

Linear independence is crucial for identifying vectors that can form bases for spaces, letting you explore different dimensions comprehensively and without redundancy.