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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

PROVE: Matrix Form Of Rotation of Axes Formulas

Let Z , Z , and R be the matrices

Z = [ x y ] Z = [ X Y ] R = [ cos ϕ sin ϕ sin ϕ cos ϕ ]

a. Show that the Rotation of Axes Formulas can be written as

Z = R Z and Z = R 1 Z

b. Let R 1 and R 2 be matrices that represent rotations through angles ϕ 1 and ϕ 2 , respectively. Show that the product matrix R 1 R 2 represents a rotation through an angle ϕ 1 + ϕ 2 [Hint: Use addition formulas for Sine and Cosine to simplify the entries of the matrix R 1 R 2 .]

To determine

(a)

To show:

That the Rotation of Axes Formulas is given as,

Z=RZ and Z=R1Z.

Explanation

Given:

Let Z, Z, and R be the matrices given as,

Z=[xy]

Z=[XY]

R=[cosϕsinϕsinϕcosϕ]

Approach:

Suppose the x and the y axes in a coordinate plane are rotated through the acute angle ϕ to produce the X and the Y axes. Then the coordinates (x,y) and (X,Y) of a point in the xy and the XY planes are related as follows,

x=(XcosϕYsinϕ)

y=(Xsinϕ+Ycosϕ)

X=(xcosϕ+ysinϕ)

Y=(xsinϕ+ycosϕ)

Calculation:

Multiply the matrices R and Z as follows,

RZ=[cosϕsinϕsinϕcosϕ][XY]

RZ=[XcosϕYsinϕXsinϕ+Ycosϕ](1)

Use x for (XcosϕYsinϕ) and y for (Xsinϕ+Ycosϕ) in equation (1) as,

RZ=[XcosϕYsinϕXsinϕ+Ycosϕ]=[xy]=Z

Therefore, Z=RZ.

The inverse of the matrix R is given as,

R1=1|R|(AdjointofR)(2)

The determinant of R is given as,

|R|=cos2ϕ+sin2ϕ=1

The Adjoint of R is given as,

To determine

(b)

The rotation of the product matrix R1R2.

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