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?

Yiming Z.

Numerade Educator

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

Alan G.

Numerade Educator

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?

Yiming Z.

Numerade Educator

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 the

coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.

Brandon T.

Numerade Educator

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

Yiming Z.

Numerade Educator

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.

Yiming Z.

Numerade Educator

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

Alan G.

Numerade Educator

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

Alan G.

Numerade Educator

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) $

Alan G.

Numerade Educator

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

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

Alan G.

Numerade Educator

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) $

Daniel B.

Numerade Educator

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

Adam D.

Numerade Educator

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?

Alan G.

Numerade Educator

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

Alan G.

Numerade Educator

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

Alan G.

Numerade Educator

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

Yiming Z.

Numerade Educator

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

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

Alan G.

Numerade Educator

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

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

Alan G.

Numerade Educator

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

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

Alan G.

Numerade Educator

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

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

Alan G.

Numerade Educator

(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.)

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

Yiming Z.

Numerade Educator

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

Sirat S.

Numerade Educator

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

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

$ x = 5 $

Yiming Z.

Numerade Educator

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

$ y = -2 $

Alan G.

Numerade Educator

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

$ y < 8 $

Alan G.

Numerade Educator

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

$ z \ge -1 $

Alan G.

Numerade Educator

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

$ 0 \le z \le 6 $

Alan G.

Numerade Educator

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

$ y^2 = 4 $

Alan G.

Numerade Educator

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

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

Alan G.

Numerade Educator

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

$ x^2 + y^2 = 4 $

Yiming Z.

Numerade Educator

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

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

Yiming Z.

Numerade Educator

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

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

Alan G.

Numerade Educator

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

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

Alan G.

Numerade Educator

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

$ x = z $

Megan S.

Numerade Educator

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

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

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

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

Isaac B.

Numerade Educator

Write inequalities to describe the region.

The region between the yz-plane and the vertical plane $ x = 5 $

Pawan Y.

Numerade Educator

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.

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

Connor P.

Numerade Educator

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

Hannah M.

Numerade Educator

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.

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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.

Grant W.

Numerade Educator

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.

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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

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

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

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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.

Janielle M.

Numerade Educator