Finding Scalar Multipliers for Vector Equation

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Homework Help Overview

The problem involves finding scalar multipliers for a vector equation, specifically determining values for a and b such that v = au + bw, where u and w are given vectors and v is a target vector.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss setting up the vector equation in component form and equating the components to form a system of equations. Some express uncertainty about the next steps after setting up the equations.

Discussion Status

There has been some productive guidance offered regarding how to set up the equations based on the vector components. Participants are exploring different interpretations of the problem setup and the subsequent steps needed to solve for a and b.

Contextual Notes

One participant notes a lack of examples in their textbook that resemble the problem, which may contribute to their uncertainty in approaching the solution.

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Homework Statement


Find a and b such that v=au + bw, where u=<1, 2> and w=<1, -1>


Homework Equations


v=au + bw
v=<2, 1>


The Attempt at a Solution


No attempt

I really don't know where to begin. There is not an example like this in the book.
 
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Well, try writing the equations in this format:
[tex]\left(\begin{array}{c}2\\1\end{array}\right)=a\left(\begin{array}{c}1\\2\end{array}\right)+b\left(\begin{array}{c}1\\-1\end{array}\right)[/tex]

Then set up equations for the x (top) component, and the y (bottom) component, and solve for a and b.
 
Your equation says that a<1, 2>+ b< 1, -1>= <2, 1> or
<a+ b, 2a- b>= <2, 1>. Since two vectors are equal only if corresponding components are equal, you have the two equations a+ b= 2, 2a- b= 1. Solve those for a and b.
 
I actually had it set up like that but thought I was doing something wrong. I didn't know what to do next. Thanks for the help!
 

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