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In Section 6.5, you are introduced to rotations of R3 by an angle about an axis through the origin. In general, the axis can be any straight line through the origin. Such rotations are linear transformations of R3 and as such have a matrix relative to the standard basis of R3. Example 4 in 6.5 discusses the matrix A for rotation of R3 counterclockwise about the z-axis by an angle of 60°. A little later, the textbook tells you how to find the matrix B for rotation about the x-axis by an angle of 30°. I want you to study the properties of the product AB. It represents the composition of two distinct rotations of R3, where first B is applied and then A is applied. We are at the stage where we could prove a wonderful theorem in linear algebra which states that the composition of such rotations of R3 is again a rotation of R3, albeit with a new axis and a new angle. But in this problem, I want you to find the axis and the angle for AB and calculate this for yourself. Step 1: Work out the matrix C = AB. Step 2: A position vector x on the axis of a rotation is left unchanged by the rotation. Hence study the equation Cx = x. Solve this equation and find a basis vector u for this eigenspace (since λ = 1 will be an eigenvalue of C). Step 3: Find all vectors v which are orthogonal to u. (You should use the ideas of dot products in section 5.1.) Step 4: Find a basis v1; v2 for the subspace of vectors v found in Step 3. Step 5: Work out Cv1 = w1 and Cv2 = w2. Step 6: Work out u · w1 and u · w2. They SHOULD be equal to zero (though due to computer rounding, they may only be very close to zero). Step 7: Find the acute angle between v1 and w1. Find the acute angle between v2 and w2. They should be the same angle, call it θ. You have now shown that AB represents the linear transformation which is rotation of R3 by the angle θ about the axis through the origin parallel to u. Visually, you have studied a rotating sphere like the earth where the axis is the line joining the north and south poles, with the vector u pointing from the center (origin) in one of those directions. The vectors v1; v2 are two vectors in the equatorial plane of the sphere and you are checking in Steps 6 and 7 that the two of them are being rotated around in that plane by the same angle. Step 8: Write up all your work. Include calculation details, explaining how you did the calculations. Your answer consists of the values you found for u and θ (the direction of the axis and the angle of rotation). Comment: I found the work easier to do helped by a computer. One possible software is MATLAB on the campus computer network (which I showed you in the beginning of the semester). A bad aspect is that I could not quite get zero in Step 6, because of rounding errors. The matrices A and B, and hence C contain entries that use √3. MATLAB replaces √3 by an approximation like 1.7321, and hence everything gets a little changed. I found a free site on the web called www.symbolab.com which does a lot of the work leaving √3 unchanged. Each had its shortcomings. But both gave me an angle of 66.45 degrees. I give this information for you as a check of your work.
Sri K.
Consider a spacecraft that is far away from planets or other massive objects. The mass of the spacecraft is M = 1.5×10^5 kg. The rocket engines are shut off and the spacecraft coasts with a velocity vector v = (0, 20, 0) km/s. The spacecraft passes the position x = (12, 15, 0) km at which time the spacecraft fires its thruster rockets giving it a net force of F = (6 × 10^4, 0, 0) N which is exerted for 3.4 s. The ejected gases have a total mass that is small compared to the mass of the spacecraft. a) Where is the spacecraft 1 hour afterwards? b) What approximations have you made in your analysis? Kepler's second law is this statement: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. We are going to prove this statement. Consider the wedge in the figure with area dA = 1/2 R^2 dθ. The rate that area is swept per unit time is dA/dt = 1/2 R^2 dθ/dt = 1/2 R^2 θ˙ and this is true even if radius R is varying. We take the origin to be the center of the Sun and radius R is the distance between the planet and the Sun. The angle θ gives the position of the planet in the ecliptic plane. Kepler's second law is equivalent to dA/dt = constant or d^2A/dt = 0. In class, we showed that acceleration in polar coordinates can be written as a = (R¨ - Rθ˙^2)r̂ + (2Ṙθ̇ + Rθ̈)θ̂. Because the gravitational force is in the radial direction, the tangential component of acceleration is zero. This means that 2Ṙθ̇ + Rθ̈ = 0. Show that this relation is equivalent to dA/dt = constant and Kepler's second law. A spherical hollow is made in a sphere of radius R = 11.3 cm such that its surface touches the outside surface of the sphere and passes through its center (see Figure). The mass of the sphere before hollowing was M = 57.0 kg. What is the magnitude of the gravitational force between the hollowed-out lead sphere and a small sphere of mass m = 4.2 kg, located a distance d = 0.55 m from the center of the lead sphere?
Geometrical optics. Consider a thin biconvex lens with two spherical faces. This is a good model for the lens of the human eye and for the lenses used in many optical instruments, such as reading glasses, cameras, microscopes, and telescopes. The line through the centers of the spheres defining the two faces is called the optical axis of the lens. In this exercise, we learn how we can track the path of a ray of light as it passes through the lens, provided that the following conditions are satisfied: $\bullet$The ray lies in a plane with the optical axis. $\bullet$The angle the ray makes with the optical axis is small. To keep track of the ray, we introduce two reference planes perpendicular to the optical axis, to the left and to the right of the lens. We can characterize the incoming ray by its slope $m$ and its intercept $x$ with the left reference plane. Likewise, we characterize the outgoing ray by slope $n$ and intercept $y$. We want to know how the outgoing ray depends on the incoming ray; that is, we are interested in the transformation \[T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} ; \quad\left[\begin{array}{l} x \\ m \end{array}\right] \rightarrow\left[\begin{array}{l} y \\ n \end{array}\right].\] We will see that $T$ can be approximated by a linear transformation provided that $m$ is small, as we assumed. To study this transformation, we divide the path of the ray into three segments, as shown in the following figure: We have introduced two auxiliary reference planes, directly to the left and to the right of the lens. Our transformation \[\left[\begin{array}{l} x \\ m \end{array}\right] \rightarrow\left[\begin{array}{l} y \\ n \end{array}\right].\] can now be represented as the composite of three $\operatorname{sim}$ pler transformations: \[\left[\begin{array}{l} x \\ m \end{array}\right] \rightarrow\left[\begin{array}{l} v \\ m \end{array}\right] \rightarrow\left[\begin{array}{l} w \\ n \end{array}\right] \rightarrow\left[\begin{array}{l} y \\ n \end{array}\right].\] From the definition of the slope of a line, we get the relations $v=x+L m$ and $y=w+R n$. It would lead us too far into physics to derive a formula for the transformation \[\left[\begin{array}{l} v \\ m \end{array}\right] \rightarrow\left[\begin{array}{l} w \\ n \end{array}\right]\] here. $^{13}$ Under the assumptions we have made, the transformation is well approximated by \[\left[\begin{array}{l} w \\ n \end{array}\right]=\left[\begin{array}{rr} 1 & 0 \\ -k & 1 \end{array}\right]\left[\begin{array}{l} v \\ m \end{array}\right],\] for some positive constant $k$ (this formula implies that $w=v)$. The transformation $\left[\begin{array}{l}x \\ m\end{array}\right] \rightarrow\left[\begin{array}{l}y \\ n\end{array}\right]$ is represented by the matrix product \[\begin{aligned} \left[\begin{array}{cc} 1 & R \\ 0 & 1 \end{array}\right]\left[\begin{array}{rr} 1 & 0 \\ -k & 1 \end{array}\right]\left[\begin{array}{cc} 1 & L \\ 0 & 1 \end{array}\right] \\ &=\left[\begin{array}{cc} 1-R k & L+R-k L R \\ -k & 1-k L \end{array}\right]. \end{aligned}\] a. Focusing parallel rays. Consider the lens in the human eye, with the retina as the right reference plane. In an adult, the distance $R$ is about 0.025 meters (about $1 \text { inch }) .$ The ciliary muscles allow you to vary the shape of the lens and thus the lens constant $k$, within a certain range. What value of $k$ enables you to focus parallel incoming rays, as shown in the figure? This value of $k$ will allow you to see a distant object clearly. (The customary unit of measurement for $\left.k \text { is } 1 \text { diopter }=\frac{1}{1 \text { meter }} .\right).$ b. What value of $k$ enables you to read this text from a distance of $L=0.3$ meters? Consider the following figure (which is not to scale). c. The telescope. An astronomical telescope consists of two lenses with the same optical axis. Find the matrix of the transformation \[\left[\begin{array}{c} x \\ m \end{array}\right] \rightarrow\left[\begin{array}{l} y \\ n \end{array}\right]. \] in terms of $k_{1}, k_{2},$ and $D .$ For given values of $k_{1}$ and $k_{2},$ how do you choose $D$ so that parallel incoming rays are converted into parallel outgoing rays? What is the relationship between $D$ and the focal lengths of the two lenses, $1 / k_{1}$ and $1 / k_{2} ?$
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The Inverse of a Linear Transformation
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