Problem 100 - Precalculus

Question

Show that \(\left\langle-\frac{2}{9} a, \frac{8}{9} b\right\rangle\) is a linear combination of \(\langle a, 3 b\rangle\) and \((-a,-b),\) for any real constants \(a\) and \(b\)

Step-by-Step Solution

Answer

Yes, it is a linear combination.

Step 1: Express the Vector as a Linear Combination

To show that \(\left\langle -\frac{2}{9} a, \frac{8}{9} b \right\rangle\) is a linear combination of \(\langle a, 3b \rangle\) and \((-a, -b)\), we need to find constants \(m\) and \(n\) such that: \[ m \langle a, 3b \rangle + n (-a, -b) = \left\langle -\frac{2}{9} a, \frac{8}{9} b \right\rangle \] Expanding both sides, this becomes: \(\left\langle ma - na, 3mb - nb \right\rangle = \left\langle -\frac{2}{9} a, \frac{8}{9} b \right\rangle\).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

Yes, it is a linear combination.

Key Concepts

Vector Operations

Vector operations are fundamental tools in mathematics, particularly in fields like physics and engineering. Understanding these operations can help in solving a variety of problems involving vectors. A vector is a quantity that has both magnitude and direction, and it can be represented in component form as \(\langle x, y \rangle\).
Some basic operations you can perform with vectors include addition, subtraction, and scalar multiplication.

• Addition: You can add two vectors by adding their corresponding components. For example, if you have vector \(\langle a_1, b_1 \rangle\) and vector \(\langle a_2, b_2 \rangle\), their sum is given by \(\langle a_1 + a_2, b_1 + b_2 \rangle\).


• Subtraction: Subtraction of vectors involves subtracting the components of one vector from another, resulting in \(\langle a_1 - a_2, b_1 - b_2 \rangle\).


• Scalar Multiplication: This involves multiplying each component of a vector by a scalar (a constant), such as multiplying vector \(\langle a, b \rangle\) by scalar \(c\) to obtain \(\langle ca, cb \rangle\).

These operations are key to manipulating vectors and finding solutions to problems that involve a combination of multiple vectors.

Constants in Linear Combinations

In mathematics, linear combinations are expressions made by multiplying vectors by constants and adding the results together. Linear combinations help to express one vector as a result of other vectors. This is particularly useful when working with vector spaces.
When dealing with linear combinations, the constants used in the expressions are crucial as they determine the scalar multiples for each vector. For instance, in the expression \(m\textbf{u} + n\textbf{v}\), \(m\) and \(n\) represent the constants which are multiplied by vectors \(\textbf{u}\) and \(\textbf{v}\) respectively.

• These constants allow vectors to be "stretched" or "shrunk" while maintaining their direction; except in cases where the constant is negative, which would reverse the vector's direction.


• By adjusting these constants, you can control how much of each vector contributes to the resultant vector.


• Finding the right constants is key to solving linear combination problems, as in our given exercise, where we needed to find constants \(m\) and \(n\) that satisfy a given equation.

Understanding how constants impact linear combinations allows for deeper insight into how vectors relate to one another within a given space.

Vectors in Mathematics

Vectors are an essential part of mathematics, providing a way to represent quantities that have both magnitude and direction. They are used extensively in various branches of mathematics as well as in physics and engineering.
Vectors can describe physical quantities like velocity, force, and displacement, which are always dependent on direction. In mathematics, vectors are usually denoted in a component form like \(\langle x, y \rangle\) in two-dimensional space, or \(\langle x, y, z \rangle\) for three dimensions.

• Vector Spaces: Vectors together with operations like addition and scalar multiplication form a vector space. This space holds all possible linear combinations of a given set of vectors.


• Applications: Vectors are crucial when solving systems of equations and creating models for understanding real-world phenomena. They aid in expressing complex movements and forces in a simplified and calculable manner.


• Geometrical Interpretation: Geometrically, vectors are represented as arrows, where the direction represents the course of the vector and the length of the arrow represents the vector's magnitude.

A solid grasp of vectors and how they interact in mathematics is crucial for solving complex problems, such as expressing one vector in terms of others through linear combinations.