Solution: 2130 Describe in words the region of R3 represented by the equations or

Suppose you start at the origin, move along the x-axis a distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?

Sketch the points s0, 5, 2d, s4, 0, 21d, s2, 4, 6d, and s1, 21, 2d on a single set of coordinate axes.

Which of the points Ps6, 2, 3d, Qs25, 21, 4d, and Rs0, 3, 8d is closest to the xz-plane? Which point lies in the yz-plane?

What are the projections of the point (2, 3, 5) on the xy-, yz-, and xz-planes? Draw a rectangular box with the origin and s2, 3, 5d as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.

Describe and sketch the surface in R3 represented by the equation x 1 y 2.

(a) What does the equation x 4 represent in R2 ? What does it represent in R3 ? Illustrate with sketches. (b) What does the equation y 3 represent in R3 ? What does z 5 represent? What does the pair of equations y 3, z 5 represent? In other words, describe the set of points sx, y, zd such that y 3 and z 5. Illustrate with a sketch.

Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? (a) Ps3, 22, 23d, Qs7, 0, 1d, Rs1, 2, 1d (b) Ps2, 21, 0d, Qs4, 1, 1d, Rs4, 25, 4d

Find the distance from s3, 7, 25d to each of the following. (a) The xy-plane (b) The yz-plane (c) The xz-plane (d) The x-axis (e) The y-axis (f) The z-axis

Determine whether the points lie on a straight line. (a) As2, 4, 2d, Bs3, 7, 22d, Cs1, 3, 3d (b) Ds0, 25, 5d, Es1, 22, 4d, Fs3, 4, 2d

Find an equation of the sphere with center s2, 26, 4d and radius 5. Describe its intersection with each of the coordinate planes.

Find an equation of the sphere that passes through the point s4, 3, 21d and has center s3, 8, 1d.

Find an equation of the sphere that passes through the origin and whose center is s1, 2, 3d

1316 Show that the equation represents a sphere, and find its center and radius. 13. x 2 1 y 2 1 z 2 2 6x 1 4y 2 2z 11 14. x 2 1 y 2 1 z 2 1 8x 2 6y 1 2z 1 17 0 15. 2x 2 1 2y 2 1 2z 2 8x 2 24z 1 1 16. 3x 2 1 3y 2 1 3z 2 10 1 6y 1 12z

1316 Show that the equation represents a sphere, and find its center and radius. 13. x 2 1 y 2 1 z 2 2 6x 1 4y 2 2z 11 14. x 2 1 y 2 1 z 2 1 8x 2 6y 1 2z 1 17 0 15. 2x 2 1 2y 2 1 2z 2 8x 2 24z 1 1 16. 3x 2 1 3y 2 1 3z 2 10 1 6y 1 12z

1316 Show that the equation represents a sphere, and find its center and radius. 13. x 2 1 y 2 1 z 2 2 6x 1 4y 2 2z 11 14. x 2 1 y 2 1 z 2 1 8x 2 6y 1 2z 1 17 0 15. 2x 2 1 2y 2 1 2z 2 8x 2 24z 1 1 16. 3x 2 1 3y 2 1 3z 2 10 1 6y 1 12z

1316 Show that the equation represents a sphere, and find its center and radius. 13. x 2 1 y 2 1 z 2 2 6x 1 4y 2 2z 11 14. x 2 1 y 2 1 z 2 1 8x 2 6y 1 2z 1 17 0 15. 2x 2 1 2y 2 1 2z 2 8x 2 24z 1 1 16. 3x 2 1 3y 2 1 3z 2 10 1 6y 1 12z

(a) Prove that the midpoint of the line segment from P1sx1, y1, z1d to P2sx2, y2, z2 d is S x1 1 x2 2 , y1 1 y2 2 , z1 1 z2 2 D (b) A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Find the lengths of the medians of the triangle with vertices As1, 2, 3d, Bs22, 0, 5d, and Cs4, 1, 5d.

. Find an equation of a sphere if one of its diameters has endpoints s2, 1, 4d and s4, 3, 10d.

Find equations of the spheres with center s2, 23, 6d that touch (a) the xy-plane, (b) the yz-plane, (c) the xz-plan

Find an equation of the largest sphere with center (5, 4, 9) that is contained in the first octant.

2130 Describe in words the region of R3 represented by the equations or inequalities. 21. x 5 22. y 22 23. y , 8 24. x > 23 25. 0 < z < 6 26. z 2 1 27. x 2 1 y 2 4, z 21 28. y 2 1 z 2 16 29. x 2 1 y 2 1 z 2 < 3 30. x z

2130 Describe in words the region of R3 represented by the equations or inequalities. 21. x 5 22. y 22 23. y , 8 24. x > 23 25. 0 < z < 6 26. z 2 1 27. x 2 1 y 2 4, z 21 28. y 2 1 z 2 16 29. x 2 1 y 2 1 z 2 < 3 30. x z

2130 Describe in words the region of R3 represented by the equations or inequalities. 21. x 5 22. y 22 23. y , 8 24. x > 23 25. 0 < z < 6 26. z 2 1 27. x 2 1 y 2 4, z 21 28. y 2 1 z 2 16 29. x 2 1 y 2 1 z 2 < 3 30. x z

2130 Describe in words the region of R3 represented by the equations or inequalities. 21. x 5 22. y 22 23. y , 8 24. x > 23 25. 0 < z < 6 26. z 2 1 27. x 2 1 y 2 4, z 21 28. y 2 1 z 2 16 29. x 2 1 y 2 1 z 2 < 3 30. x z

2130 Describe in words the region of R3 represented by the equations or inequalities. 21. x 5 22. y 22 23. y , 8 24. x > 23 25. 0 < z < 6 26. z 2 1 27. x 2 1 y 2 4, z 21 28. y 2 1 z 2 16 29. x 2 1 y 2 1 z 2 < 3 30. x z

2130 Describe in words the region of R3 represented by the equations or inequalities. 21. x 5 22. y 22 23. y , 8 24. x > 23 25. 0 < z < 6 26. z 2 1 27. x 2 1 y 2 4, z 21 28. y 2 1 z 2 16 29. x 2 1 y 2 1 z 2 < 3 30. x z

2130 Describe in words the region of R3 represented by the equations or inequalities. 21. x 5 22. y 22 23. y , 8 24. x > 23 25. 0 < z < 6 26. z 2 1 27. x 2 1 y 2 4, z 21 28. y 2 1 z 2 16 29. x 2 1 y 2 1 z 2 < 3 30. x z

2130 Describe in words the region of R3 represented by the equations or inequalities. 21. x 5 22. y 22 23. y , 8 24. x > 23 25. 0 < z < 6 26. z 2 1 27. x 2 1 y 2 4, z 21 28. y 2 1 z 2 16 29. x 2 1 y 2 1 z 2 < 3 30. x z

2130 Describe in words the region of R3 represented by the equations or inequalities. 21. x 5 22. y 22 23. y , 8 24. x > 23 25. 0 < z < 6 26. z 2 1 27. x 2 1 y 2 4, z 21 28. y 2 1 z 2 16 29. x 2 1 y 2 1 z 2 < 3 30. x z

2130 Describe in words the region of R3 represented by the equations or inequalities. 21. x 5 22. y 22 23. y , 8 24. x > 23 25. 0 < z < 6 26. z 2 1 27. x 2 1 y 2 4, z 21 28. y 2 1 z 2 16 29. x 2 1 y 2 1 z 2 < 3 30. x z

3134 Write inequalities to describe the region. 31. The region between the yz-plane and the vertical plane x 5 32. The solid cylinder that lies on or below the plane z 8 and on or above the disk in the xy-plane with center the origin and radius 2

3134 Write inequalities to describe the region. 31. The region between the yz-plane and the vertical plane x 5 32. The solid cylinder that lies on or below the plane z 8 and on or above the disk in the xy-plane with center the origin and radius 2

3134 Write inequalities to describe the region. 31. The region between the yz-plane and the vertical plane x 5 32. The solid cylinder that lies on or below the plane z 8 and on or above the disk in the xy-plane with center the origin and radius 2 3. The region consisting of all points between (but not on) the spheres of radius r and R centered at the origin, where r , R 34. The solid upper hemisphere of the sphere of radius 2 centered at the origin

3134 Write inequalities to describe the region. 31. The region between the yz-plane and the vertical plane x 5 32. The solid cylinder that lies on or below the plane z 8 and on or above the disk in the xy-plane with center the origin and radius 2 3. The region consisting of all points between (but not on) the spheres of radius r and R centered at the origin, where r , R 34. The solid upper hemisphere of the sphere of radius 2 centered at the origin

The figure shows a line L1 in space and a second line L2, which is the projection of L1 onto the xy-plane. (In other words, the points on L2 are directly beneath, or above, the points on L1.) (a) Find the coordinates of the point P on the line L1. (b) Locate on the diagram the points A, B, and C, where the line L1 intersects the xy-plane, the yz-plane, and the xz-plane, respectively.

Darwins finches have been used to study how differences in bird morphology are related to differences in diet. Morphological measurements (in mm) of three species are given in the table for three traits. Species Wing length Tarsus length Beak length G. difficilis 64 18.1 9.6 G. fuliginosa 62.1 17.9 8.6 G. scandens 73.1 21.1 14.5 The proportion of time spent feeding on different types of food for these three species is given in the following table. Species Seeds Pollen Other G. difficilis 0.67 0.23 0.1 G. fuliginosa 0.7 0.28 0.02 G. scandens 0.14 0 0.86 (a) Thinking of the morphology of each species as a point in R3 , calculate the morphological distance between each pair of species. (b) Thinking of the diet of each species as a point in R3 , calculate the diet distance between each pair of species

Human biomechanics The trajectory of the center of a human walking on a treadmill is shown in the figure. Here x denotes the lateral position, y the position forward or backward, and z the vertical position. The coordinates at time t 0 are s1.2, 0, 1.5d and at t 2 are s1, 2, 0.5d, where distances are measured in cm. (a) What is magnitude of the the net lateral distance traveled over the first two seconds? (b) What is the magnitude of the net distance traveled in the vertical direction over the first two seconds? (c) What is the net distance traveled through three-dimensional space over the first two seconds?

Vaccine design Most vaccines protect only against pathogens that fall within a certain region of antigenic space. Suppose that a vaccine protects against any strain falling within a sphere of radius 2 centered at the point s2, 1, 0d in antigenic space. For each strain, determine whether this vaccine will be effective. (a) A strain located at s0, 0, 0d in antigenic space (b) A strain located at s1, 0, 3d in antigenic space (c) A strain located at s1, 0, 1d in antigenic space (d) A strain located at s1y4, 2, 1d in antigenic space

Antigenic evolution and vaccination Antigenic data like those in Figure 14 can be summarized by taking the points from each year and drawing the smallest possible circle that encompasses these data. This results in a temporal sequence of circles in antigenic space, one for each ear. Similarly, if the antigenic data are in three dimensions, spheres can be drawn around the data for each year, as shown in the figure. If the circles (or spheres) from years x and x 1 1 overlap, then the amount of antigenic change between these years is relatively small. In such cases we might expect a single vaccine to work for both years. If the circles or spheres do not overlap, then we might need different vaccines for each year. z y x 2010 2011 2012 2013 (a) Suppose the antigenic data are two-dimensional, and the circles for two successive years are given by the equations sx 2 2d 2 1 s y 2 3d 2 1 and sx 2 3d 2 1 s y 2 2d 2 1 4. Would a single vaccine work for both years? (b) Suppose the antigenic data are three-dimensional, and the spheres for two successive years are given by the equations sx 2 2d 2 1 s y 2 3d 2 1 sz 2 1d 2 1 and sx 2 3d 2 1 s y 2 2d 2 1 z 2 1 4 Would a single vaccine work for both years?

Describe and sketch a solid with the following properties: When illuminated by rays parallel to the z-axis, its shadow is a circular disk. If the rays are parallel to the y-axis, its shadow is a square. If the rays are parallel to the x-axis, its shadow is an isosceles triangle.