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Chapter 1: Problem 22

Draw the standard coordinate axes on the same diagram as the axes relative to u and v. Use these to find \(\mathbf{w}\) as a linear combination of u and v. $$\mathbf{u}=\left[\begin{array}{r} -2 \\ 3 \end{array}\right], \mathbf{v}=\left[\begin{array}{l} 2 \\ 1 \end{array}\right], \mathbf{w}=\left[\begin{array}{l} 2 \\ 9 \end{array}\right]$$

Short Answer

Expert verified
\( \mathbf{w} = 2\mathbf{u} + 3\mathbf{v} \).

Step by step solution

01

Plot Standard Coordinate Axes

On a graph, draw the standard coordinate axes. The x-axis is horizontal, and the y-axis is vertical. These will serve as reference for plotting vectors.
02

Plot Vectors u and v

Plot vector \( \mathbf{u} = \begin{bmatrix} -2 \ 3 \end{bmatrix} \) by starting at the origin, moving 2 units left (negative direction of the x-axis) and 3 units up. Next, plot vector \( \mathbf{v} = \begin{bmatrix} 2 \ 1 \end{bmatrix} \) by starting at the origin, moving 2 units right and 1 unit up.
03

Establish u-v Coordinate System

Treat vectors \( \mathbf{u} \) and \( \mathbf{v} \) as the new coordinate axes. This system allows us to express any vector as a combination of \( \mathbf{u} \) and \( \mathbf{v} \).
04

Express Vector w as a Linear Combination of u and v

To express \( \mathbf{w} = \begin{bmatrix} 2 \ 9 \end{bmatrix} \) as a linear combination of \( \mathbf{u} \) and \( \mathbf{v} \), solve for scalars \( c_1 \) and \( c_2 \) such that \( \mathbf{w} = c_1 \mathbf{u} + c_2 \mathbf{v} \). This translates to solving the system:\[-2c_1 + 2c_2 = 2 \3c_1 + c_2 = 9\]
05

Solve the System for c_1 and c_2

Solve the system of equations:1. From \( -2c_1 + 2c_2 = 2 \), simplify to \( -c_1 + c_2 = 1 \) by dividing through by 2.2. Use substitution or elimination to find:\[3c_1 + c_2 = 9 \Rightarrow c_2 = 9 - 3c_1 \-c_1 + (9 - 3c_1) = 1 \Rightarrow 4c_1 = 8 \Rightarrow c_1 = 2\]Substitute \( c_1 = 2 \) into \( c_2 = 9 - 3c_1 \) to get \( c_2 = 3 \).
06

Confirm Vector w as Linear Combination

Verify the solution by substituting \( c_1 \) and \( c_2 \) back into \( \mathbf{w} = c_1 \mathbf{u} + c_2 \mathbf{v} \):\[\mathbf{w} = 2 \begin{bmatrix} -2 \ 3 \end{bmatrix} + 3 \begin{bmatrix} 2 \ 1 \end{bmatrix} = \begin{bmatrix} 2 \ 9 \end{bmatrix}\]This confirms that \( \mathbf{w} = 2\mathbf{u} + 3\mathbf{v}. \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Vectors
In mathematics and physics, vectors are quantities that have both magnitude and direction. Think of them as arrows pointing in a particular direction, with a specific length. Vectors are often used to represent forces, velocities, and many other physical quantities.
  • Each vector can be represented as a pair or triplet of numbers, denoting its position relative to an origin. For instance, the vector \( \mathbf{u} = \begin{bmatrix} -2 \ 3 \end{bmatrix} \) represents a movement 2 units left and 3 units upward from the origin.
  • Vectors can be added together to form new vectors. The sum
    \( \mathbf{u} + \mathbf{v} \) creates a new vector that combines the effects (directions and magnitudes) of both \( \mathbf{u} \) and \( \mathbf{v} \).
  • Vectors can be scaled by multiplying them with a constant, known as a scalar. This changes their length without altering their direction.
Understanding how vectors work is crucial for solving problems involving positions, velocities, and other vector quantities. They provide a way to simplify many physical and mathematical phenomena, making complex problems easier to visualize and solve.
Coordinate Systems and Transformations
A coordinate system allows us to represent points and vectors in space using a set of numbers, usually within a graph.
  • The most common is the Cartesian coordinate system, where positions are defined by \(x\) (horizontal axis) and \(y\) (vertical axis) values.
  • In our exercise, vectors \(\mathbf{u}\) and \(\mathbf{v}\) create a new coordinate system. This u-v coordinate system provides a unique perspective for evaluating vectors. Here, any point or vector can be described using linear combinations of \(\mathbf{u}\) and \(\mathbf{v}\), rather than traditional \(x\) and \(y\) axes.
Coordinate transformations are used to switch between different coordinate systems, making it easier to solve problems in various applications, such as geometry, physics, and engineering.
Solving a System of Equations
A system of equations involves finding values for variables that satisfy all equations in the system. When dealing with vectors, systems of equations can be used to express a particular vector as a linear combination of other vectors.
  • In our example, the problem involves expressing vector \( \mathbf{w} \) as \( c_1 \mathbf{u} + c_2 \mathbf{v} \) by finding the appropriate values of scalars \( c_1 \) and \( c_2 \).
  • Such systems are solved using various methods like substitution, elimination, or matrix operations.
  • The goal is to find values for the scalar multipliers that make the linear combination equal to the target vector, \( \mathbf{w} \).
These methods are not only fundamental in algebra, but they are also handy in determining relationships between variables in many fields including computer graphics, physics, and economics. By solving a system of equations, we can uncover hidden insights and connections between different mathematical elements.

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Most popular questions from this chapter

Solve the given equation or indicate that there is no solution. $$6 x=5 \text { in } \mathbb{Z}_{8}$$

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Draw the standard coordinate axes on the same diagram as the axes relative to u and v. Use these to find \(\mathbf{w}\) as a linear combination of u and v. $$\mathbf{u}=\left[\begin{array}{r} 1 \\ -1 \end{array}\right], \mathbf{v}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], \mathbf{w}=\left[\begin{array}{l} 2 \\ 6 \end{array}\right]$$

Determine whether the angle between \(\mathbf{u}\) and \(\mathbf{v}\) is acute, obtuse, or a right angle. $$\mathbf{u}=\left[\begin{array}{r} 2 \\ -1 \\ 1 \end{array}\right], \mathbf{v}=\left[\begin{array}{r} 1 \\ -2 \\ -1 \end{array}\right]$$
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