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Section 1

Three-Dimensional Coordinate Systems

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 $ (1, 5, 3) $, $ (0, 2, -3) $, $ (-3, 0, 2) $, and $ (2, -2, -1) $ on a single set of coordinate axes.

Which of the points $ A (-4, 0, -1) $, $ B (3, 1, -5) $, and $ C (2, 4, 6) $ is closest to the $ yz $-plane? Which point lies in the $ xz $-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 $ (2, 3, 5) $ as opposite vertices and with its faces parallel to thecoordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.

What does the equation $ x = 4 $ represent in $ \mathbb{R}^2 $? What does it represent in $ \mathbb{R}^3 $? Illustrate with sketches.

What does the equation $ y = 3 $ represent in $ \mathbb{R}^3 $? What does $ z = 5 $ represent? What does the pair of equations $ y = 3 $, $ z = 5 $ represent? In other words, describe the set of points $ (x, y, z) $ such that $ y = 3 $ and $ z = 5 $. Illustrate with a sketch.

Describe and sketch the surface in $ \mathbb{R}^3 $ represented by the equation $ x + y = 2 $.

Describe and sketch the surface $ \mathbb{R}^3 $ represented by the equation $ x^2 + z^2 = 9 $.

Find the lengths of the sides of the triangle $ PQR $. Is it a right triangle? Is it an isosceles triangle?

$ P (3, -2, -3) $ , $ Q (7, 0, 1) $ , $ R (1, 2, 1) $

$ P (2, -1, 0) $ , $ Q (4, 1, 1) $ , $ R (4, -5, 4) $

Determine whether the points lie on a straight line.

(a) $ A (2, 4, 2) $ , $ B (3, 7, -2) $, $ C (1, 3, 3) $(b) $ D (0, -5, 5) $ , $ E (1, -2, 4) $, $ F (3, 4, 2) $

Find the distance from $ (4, -2, 6) $ 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

Find an equation of the sphere with center $ (-3, 2, 5) $ and radius 4. What is the intersection of this sphere with the $ yz $-plane?

Find an equation of the sphere with center $ (2, -6, 4) $ and radius 5. Describe its intersection with each of the coordinate planes.

Find an equation of the sphere that passes through the point $ (4, 3, -1) $ and has center $ (3, 8, 1) $.

Find an equation of the sphere that passes through the origin and whose center is $ (1, 2, 3) $.

Show that the equation represents a sphere, and find its center and radius.

$ x^2 + y^2 + z^2 - 2x - 4y + 8z = 15 $

$ x^2 + y^2 + z^2 + 8x - 6y + 2z + 17 = 0 $

$ 2x^2 + 2y^2 + 2z^2 = 8x - 24z + 1 $

$ 3x^2 + 3y^2 + 3z^2 = 10 + 6y + 12z $

(a) Prove that the midpoint of the line segment from $ P_1 (x_1, y_1, z_1) $ to $ P_2 (x_2, y_2, z_2) $ is$$ \left(\frac{x_1 + x_2}{2} , \frac{y_1+ y_2}{2} , \frac{z_1+ z_2}{2} \right) $$(b) Find the lengths of the medians of the triangle with vertices $ A (1, 2, 3) $, $ B (-2, 0, 5) $, $ C (4, 1, 5) $. (A median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side.)

Find an equation of a sphere if one of its diameters has endpoints $ (5, 4, 3) $ and $ (1, 6, -9) $.

Find equations of the spheres with center $ (2, -3, 6) $ that touch (a) the $ xy $-plane, (b) the $ yz $-plane, (c) the $ xz $-plane.

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

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ x = 5 $

$ y = -2 $

$ y < 8 $

$ z \ge -1 $

$ 0 \le z \le 6 $

$ y^2 = 4 $

$ x^2 + y^2 = 4 $, $ z = -1 $

$ x^2 + y^2 = 4 $

$ x^2 + y^2 + z^2 = 4 $

$ x^2 + y^2 + z^2 \le 4 $

$ 1 \le x^2 + y^2 + z^2 \le 5 $

$ x = z $

$ x^2 + z^2 \le 9 $

$ x^2 + y^2 + z^2 > 2z $

Write inequalities to describe the region.The region between the yz-plane and the vertical plane $ x = 5 $

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.

The region consisting of all points between (but not on) the spheres of radius $ r $ and $ R $ centered at the origin, where $ r < R $.

The solid upper hemisphere of the sphere of radius 2 centered at the origin

The figure shows a line $ L_1 $ in space and a second line $ L_2 $, which is the projection of $ L_1 $ onto the $ xy $-plane. (In other words, the points on $ L_2 $ are directly beneath, or above, the points on $ L_1 $.)(a) Find the coordinates of the point $ P $ on the line $ L_1 $.(b) Locate on the diagram the points $ A $, $ B $, and $ C $, where the line $ L_1 $ intersects the $ xy $-plane, the $ yz $-plane, and the $ xz $-plane, respectively.

Consider the points $ P $ such that the distance from $ P $ to $ A (-1, 5, 3) $ is twice the distance from $ P $ to $ B (6, 2, -2) $. Show that the set of all such points is a sphere, and find its center and radius.

Find an equation of the set of all points equidistant from the points $ A (-1, 5, 3) $ and $ B (6, 2, -2) $. Describe the set.

Find the volume of the solid that lies inside both of the spheres$$ x^2 + y^2 + z^2 + 4x -2y + 4z + 5 = 0 $$and$$ x^2 + y^2 + z^2 = 4 $$

Find the distance between the spheres $ x^2 + y^2 + z^2 = 4 $ and $ x^2 + y^2 + z^2 = 4x + 4y + 4z - 11 $.

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.