8. A theme that will later unfold concerns the use of coordinatesystems. We can identify the point (x, y) with the tip of the vectordrawn emanating from the origin. We can then think of the usualCartesian coordinate system in terms of linear combinations of thevectorsej =e2 =V2V1(2, –3)42,–3}Figure 2.1.8. The usual Cartesian coordinate system, defined by thevectors ej and e2, is shown on the left along with the representation ofthe point (2, -3). The right shows a nonstandard coordinate systemdefined by vectors v1 and v2.The point (2, -3) is identified with the vector— 2е1 — Зе2.If we have vectorsv1 =,v2 =we may define a new coordinate system, such that a point {x, y} willcorrespond to the vectorxv1 + yv2.For instance, the point {2, –3} is shown on the right side of Figure 2.1.8Write the point {2, –3} in standard coordinates; that is, find r andy such thatа.(», у) 3 {2, —3}.b. Write the point (2, –3) in the new coordinate system; that is, finda and b such that{а, b} 3 (2, —3).Convert a general point {a, b}, expressed in the new coordinatesystem, into standard Cartesian coordinates (x, y).С.d. What is the general strategy for converting a point from standardCartesian coordinates (x, y) to the new coordinates {a, b}?Actually implementing this strategy in general may take a bit ofwork so just describe the strategy. We will study this in moredetail later.

Answered: 8. A theme that will later unfold…

Algebra and Trigonometry (6th Edition)

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Author:Robert F. Blitzer

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ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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8. A theme that will later unfold concerns the use of coordinate
systems. We can identify the point (x, y) with the tip of the vector
drawn emanating from the origin. We can then think of the usual
Cartesian coordinate system in terms of linear combinations of the
vectors
ej =
e2 =
V2
V1
(2, –3)
42,–3}
Figure 2.1.8. The usual Cartesian coordinate system, defined by the
vectors ej and e2, is shown on the left along with the representation of
the point (2, -3). The right shows a nonstandard coordinate system
defined by vectors v1 and v2.
The point (2, -3) is identified with the vector
— 2е1 — Зе2.
If we have vectors
v1 =
,v2 =
we may define a new coordinate system, such that a point {x, y} will
correspond to the vector
xv1 + yv2.
For instance, the point {2, –3} is shown on the right side of Figure 2.1.8
Write the point {2, –3} in standard coordinates; that is, find r and
y such that
а.
(», у) 3 {2, —3}.
b. Write the point (2, –3) in the new coordinate system; that is, find
a and b such that
{а, b} 3 (2, —3).
Convert a general point {a, b}, expressed in the new coordinate
system, into standard Cartesian coordinates (x, y).
С.
d. What is the general strategy for converting a point from standard
Cartesian coordinates (x, y) to the new coordinates {a, b}?
Actually implementing this strategy in general may take a bit of
work so just describe the strategy. We will study this in more
detail later.

Expert Solution

Step 1: Finding (x,y) for given {a,b} i.e. {2,-3}

Note: There are multiple sub questions. No specific question has been asked. So, We will solve the first three questions for you. For other questions, you can ask those as a new query.

$A n y v e c t o r i n o r i g i n a l s y s t e m c a n b e r e p r e s e n t e d u \sin g n e w s y s t e m a s f o l l o w s (\begin{matrix} x \\ y \end{matrix}) = a v_{1} + b v_{2} (\begin{matrix} x \\ y \end{matrix}) = a [\begin{matrix} 2 \\ 1 \end{matrix}] + b [\begin{matrix} 1 \\ 2 \end{matrix}] (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 2 a + b \\ a + 2 b \end{matrix}) G i v e n, \{a, b\} = \{2, - 3\} H e n c e (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 2 * 2 + (- 3) \\ 2 + 2 (- 3) \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 1 \\ - 4 \end{matrix})$

(1,-4) = {2,-3}

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