Question

Show that any vector \(\mathbf{V}\) in a plane can be written as a linear combination of two non-parallel 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 \(\mathbf{A} \times \mathbf{V}\) and \(\mathbf{B} \times \mathbf{V} ;\) what are \(\mathbf{A} \times \mathbf{A}\) and \(\mathbf{B} \times \mathbf{B} ?\) Take components perpendicular to the plane to show that $$a=\frac{(\mathbf{B} \times \mathbf{V}) \cdot \mathbf{n}}{(\mathbf{B} \times \mathbf{A}) \cdot \mathbf{n}}$$ 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 \(a \textbf{A} + b \textbf{B}\) where \[a = \frac{(\textbf{B} \times \textbf{V}) \cdot \textbf{n}}{(\textbf{B} \times \textbf{A}) \cdot \textbf{n}}\] and \[b = \frac{(\textbf{V} \times \textbf{A}) \cdot \textbf{n}}{(\textbf{B} \times \textbf{A}) \cdot \textbf{n}}\].

Step-by-step solution

Understand the Problem

Given vectors \(\textbf{V}\), \(\textbf{A}\), and \(\textbf{B}\) in a plane with \(\textbf{A}\) and \(\textbf{B}\) as non-parallel vectors. The goal is to express \(\textbf{V}\) as a linear combination of \(\textbf{A}\) and \(\textbf{B}\); that is, to find scalars \(a\) and \(b\) such that \(\textbf{V}=a \textbf{A} + b \textbf{B}\).

Explore Cross Products

Find the cross products involving \(\textbf{A}\) and \(\textbf{B}\): \(\textbf{A} \times \textbf{V}\) and \(\textbf{B} \times \textbf{V}\). These will help in isolating the perpendicular components to the plane.

Analyze the Given Vectors

Think about the cross products \(\textbf{A} \times \textbf{A}\) and \(\textbf{B} \times \textbf{B}\). Since any vector crossed with itself is the zero vector, \(\textbf{A} \times \textbf{A} = \textbf{0}\) and \(\textbf{B} \times \textbf{B} = \textbf{0}\).

Use Normal Vector \(\textbf{n}\)

Define \(\textbf{n}\) as a vector normal to the plane containing \(\textbf{A}\) and \(\textbf{B}\). Then, consider the components in the direction of the normal vector \(\textbf{n}\).

Set Up the Equations

Derive the equations for the scalar coefficients \(a\) and \(b\): \[a = \frac{(\textbf{B} \times \textbf{V}) \cdot \textbf{n}}{(\textbf{B} \times \textbf{A}) \cdot \textbf{n}}\] and similarly for \(b\).

Solve for the Scalars

Given the derived equations, compute \(a\) and \(b\) by substituting the necessary cross products and dot products with the normal vector \(\textbf{n}\).

Verify the Combination

Check that the computed values of \(a\) and \(b\) satisfy \(\textbf{V} = a \textbf{A} + b \textbf{B}\) and ensure all operations comply with vector properties.

Key concepts

Cross Product

The cross product of two vectors, denoted as \(\textbf{A} \times \textbf{B}\), is a vector that is perpendicular to both \(\textbf{A}\) and \(\textbf{B}\). To find the cross product, you can use the determinant of a matrix that includes the unit vectors \(\hat{i}, \hat{j}, \hat{k}\) and the components of \(\textbf{A}\) and \(\textbf{B}\). Here's the general form:
\[ \textbf{A} \times \textbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} \]
This results in a vector with three components. The direction of the resulting vector follows the right-hand rule.
The cross product is useful because it helps determine the perpendicular components of vectors, which is vital in solving the given exercise.

Normal Vector

A normal vector \(\textbf{n}\) is perpendicular to a given plane. For vectors in a 2D plane, the normal vector is commonly found using the cross product of two vectors in the plane. If we have vectors \(\textbf{A}\) and \(\textbf{B}\), their cross product \(\textbf{A} \times \textbf{B}\) will result in a normal vector \(\textbf{n}\).
In the context of the exercise, the normal vector is defined as the component that helps in isolating the perpendicular components during calculations. For instance, it's employed in evaluating the cross products and dot products accurately.
When calculating the coefficient \(a\) in the expression \(\textbf{V} = a\textbf{A} + b\textbf{B}\), the formula is: \[a = \frac{(\textbf{B} \times \textbf{V}) \cdot \textbf{n}}{(\textbf{B} \times \textbf{A}) \cdot \textbf{n}} \]
This adjustment ensures that our solution considers perpendicular components in deriving the linear combination coefficients.

Dot Product

The dot product, also known as the scalar product, is an operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. Algebraically, the dot product of two vectors \(\textbf{A}\) and \(\textbf{B}\) is given by:
\[ \textbf{A} \cdot \textbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \]
For vectors \(\textbf{A} = (a_1, a_2, a_3)\) and \(\textbf{B} = (b_1, b_2, b_3)\). The result is a scalar value.
Geometrically, the dot product measures the cosine of the angle between two vectors scaled by the magnitudes of the vectors. \[ \textbf{A} \cdot \textbf{B} = |\textbf{A}| |\textbf{B}| \cos(\theta) \]
In the exercise, the dot product helps provide insight into the perpendicular components of the vectors when combined with the cross products, especially when working with the normal vector \(\textbf{n}\). This is evident in the formula for \(a\) and \(b\) where it involves normalization based on perpendicular directional analysis.