Question

Show that any vector \(V\) in a plane can be written as a linear combination of two nonparallel vectors \(\mathbf{A}\) and \(\mathbf{B}\) in the plane; that is, find \(a\) and \(b\) so that \(\mathbf{V}=a \mathbf{A}+b \mathbf{B}\). Hint; Find the cross products \(A \times V\) and \(B \times V ;\) what are \(A \times A\) and \(B \times\) B? Take components perpendicular to the plane to show that $$ a=\frac{(B \times \mathbf{V}) \cdot \mathbf{n}}{(B \times A) \cdot n} $$ where \(\mathbf{n}\) is normal to the plane, and a similar formula for \(b\).

Short answer

Any vector \(\textbf{V}\) in the plane can be written as a linear combination of \ a \textbf{A} + b \textbf{B} \. Coefficients are derived using cross products: \[ a = \frac{(\textbf{B} \times \textbf{V}) \bullet \textbf{n}}{(\textbf{B} \times \textbf{A}) \bullet \textbf{n}}, \ b = \frac{(\textbf{A} \times \textbf{V}) \bullet \textbf{n}}{(\textbf{A} \times \textbf{B}) \bullet \textbf{n}} \]

Step-by-step solution

Define the given vectors and the unknowns

Let \(\textbf{A} = (a_1, a_2, a_3)\), \(\textbf{B} = (b_1, b_2, b_3)\), and \(\textbf{V} = (v_1, v_2, v_3)\). Assume \(\textbf{n}\) is perpendicular to the plane containing these vectors.

Calculate cross products

Compute the cross products: \(\textbf{A} \times \textbf{B}\), \(\textbf{A} \times \textbf{V}\), and \(\textbf{B} \times \textbf{V}\). The results should correspond to vectors perpendicular to \(\textbf{A}\), \(\textbf{B}\), and \(\textbf{V}\).

Express orthogonal conditions

Since the cross product results are normal to the plane, express \(a \textbf{A} + b \textbf{B} = \textbf{V}\) in terms of their vector components orthogonal to \(\textbf{A}\) and \(\textbf{B}\).

Isolate the coefficients

Use the provided hint expressions: \[ a = \frac{(\textbf{B} \times \textbf{V}) \bullet \textbf{n}}{(\textbf{B} \times \textbf{A}) \bullet \textbf{n}} \] and \[ b = \frac{(\textbf{A} \times \textbf{V}) \bullet \textbf{n}}{(\textbf{A} \times \textbf{B}) \bullet \textbf{n}} \] to solve for \ a \ and \ b \.

Summarize the relationships

Confirm the derived equations are valid for the plane by plugging the values of \ a \ and \ b \ back into \(a \textbf{A} + b \textbf{B} = \textbf{V}\)

Key concepts

Cross Product

The cross product is a binary operation on two vectors in three-dimensional space. It produces a third vector that is perpendicular to both original vectors. The magnitude of the cross product corresponds to the area of the parallelogram that the vectors span. This is especially useful in determining orthogonal vectors.
Formula: If \[\textbf{A} = (a_1, a_2, a_3)\] and \[\textbf{B} = (b_1, b_2, b_3)\], then \[\textbf{A} \times \textbf{B} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)\].
In this exercise, the cross products \[\textbf{A} \times \textbf{V}\] and \[\textbf{B} \times \textbf{V}\] give us vectors that are perpendicular to the plane of \[\textbf{A}\], \[\textbf{B}\], and \[\textbf{V}\]. This orthogonality is key to deriving the coefficients for the linear combination.

Vector Spaces

A vector space is a fundamental structure in linear algebra. It is a collection of vectors that can be added together and multiplied by scalars to produce another vector in the space. The vectors \[\textbf{A}\] and \[\textbf{B}\] in the plane form a basis if they are linearly independent.
In our exercise, the plane defined by the vectors \[\textbf{A}\] and \[\textbf{B}\] is a vector space. Every vector \[\textbf{V}\] in this plane can be represented as a linear combination of \[\textbf{A}\] and \[\textbf{B}\]. This means that we can find scalars \[a\] and \[b\] such that \[\textbf{V} = a \textbf{A} + b \textbf{B}\].

Orthogonality

Orthogonality refers to the perpendicularity of vectors. Two vectors are orthogonal if their dot product is zero. In our case, the cross products \[\textbf{A} \times \textbf{V}\] and \[\textbf{B} \times \textbf{V}\] are orthogonal (perpendicular) to the plane of the vectors.
This property helps in isolating the coefficients \[a\] and \[b\] because when you take the dot product of an orthogonal vector with any vector in the plane, the result clearly simplifies the calculations. This step is crucial in achieving the expression of vector \[\textbf{V}\] in terms of \[\textbf{A}\] and \[\textbf{B}\].

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It is fundamental in understanding vector decomposition and combinations.
In this exercise, we use linear algebra techniques to solve for the coefficients \[a\] and \[b\] in the expression \[ \textbf{V} = a \textbf{A} + b \textbf{B} \]. Techniques include cross products and properties of orthogonality in vector spaces.

Vector Decomposition

Vector decomposition is the process of breaking a vector down into its component parts. In a plane, any vector can be expressed as a linear combination of two non-parallel basis vectors.
This concept is crucial for our exercise. By finding the appropriate coefficients \[a\] and \[b\], we effectively decompose vector \[\textbf{V}\] into components along vectors \[\textbf{A} \] and \[ \textbf{B}\].
The provided hints guide us to use cross products and the normal vector \[\textbf{n}\] to achieve this. Specifically, we use: \[ a = \frac{(\textbf{B} \times \textbf{V}) \cdot \textbf{n}}{(\textbf{B} \times \textbf{A}) \cdot \textbf{n}} \] and \[ b = \frac{(\textbf{A} \times \textbf{V}) \cdot \textbf{n}}{(\textbf{A} \times \textbf{B}) \cdot \textbf{n}} \] to find our answers.