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8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285740621

Chapter 12.6, Problem 15E

Textbook Problem

**11–20** Use traces to sketch and identify the surface.

Expert Solution

To determine

**To use:**

The traces to sketch and identify the surface

**1) Concept:**

To sketch a graph of the surface, determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces.

Plot a graph of the equation

**2) Given:**

The equation

**3) Calculation:**

The given equation is

Now, the traces in the plane

Calculus (MindTap Course List)

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Ch. 12.1 - Suppose you start at the origin, move along the...Ch. 12.1 - Sketch the points (1,5,3),(0,2,3),(3,0,2), and...Ch. 12.1 - Which of the points A(4,0,1),B(3,1,5), and...Ch. 12.1 - What are the projections of the point 2, 3, 5 on...Ch. 12.1 - What does the equation x=4 represent in 2? What...Ch. 12.1 - What does the equation y = 3 represent in 3? What...Ch. 12.1 - Describe and sketch the surface in 3 represented...Ch. 12.1 - Describe and sketch the surface in 3 represented...Ch. 12.1 - Find the lengths of the sides of the triangle PQR....Ch. 12.1 - Find the lengths of the sides of the triangle PQR....

Ch. 12.1 - Determine whether the points lie on a straight...Ch. 12.1 - Find the distance from (4,2,6) to each of the...Ch. 12.1 - Find an equation of the sphere with centre (3,2,5)...Ch. 12.1 - Find an equation of the sphere with centre (2,6,4)...Ch. 12.1 - Find an equation of the sphere that passes through...Ch. 12.1 - Find an equation of the sphere that passes through...Ch. 12.1 - Show that the equation represents a sphere, and...Ch. 12.1 - Show that the equation represents a sphere, and...Ch. 12.1 - Show that the equation represents a sphere, and...Ch. 12.1 - Show that the equation represents a sphere, and...Ch. 12.1 - a Prove that the midpoint of the line segment from...Ch. 12.1 - Find an equation of a sphere if one of its...Ch. 12.1 - Find equations of the spheres with centre (2,3,6)...Ch. 12.1 - Find an equation of the largest sphere with centre...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Write inequalities to describe the region. The...Ch. 12.1 - Write inequalities to describe the region. The...Ch. 12.1 - Write inequalities to describe the region. The...Ch. 12.1 - Write inequalities to describe the region. The...Ch. 12.1 - The figure shows a line L1 in space and a second...Ch. 12.1 - Consider the points P such that the distance from...Ch. 12.1 - Find an equation of the set of all points...Ch. 12.1 - Find the volume of the solid that lies inside both...Ch. 12.1 - Find the distance between the spheres x2+y2+z2=4...Ch. 12.1 - Describe and sketch a solid with the following...Ch. 12.2 - Are the following quantities vectors or scalars?...Ch. 12.2 - What is the relationship between the point 4, 7...Ch. 12.2 - Name all the equal vectors in the parallelogram...Ch. 12.2 - Write each combination of vectors as a single...Ch. 12.2 - Copy the vectors in the figure and use them to...Ch. 12.2 - Copy the vectors in the figure and use them to...Ch. 12.2 - In the figure, the tip of c and the tail of d are...Ch. 12.2 - If the vectors in the figure satisfy |u|=|v|=1 and...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Find the sum of the given vectors and illustrate...Ch. 12.2 - Find the sum of the given vectors and illustrate...Ch. 12.2 - Find the sum of the given vectors and illustrate...Ch. 12.2 - Find the sum of the given vectors and illustrate...Ch. 12.2 - Find a+b,4a+2b,|a|, and |ab|. a=3,4,b=9,1Ch. 12.2 - Find a+b,4a+2b,|a|, and |ab|. a=5i+3j,b=i2jCh. 12.2 - Find a+b,4a+2b,|a|, and |ab|. a=4i3j+2k,b=2i4kCh. 12.2 - Find a+b,4a+2b,|a|, and |ab|. a=8,1,4,b=5,2,1Ch. 12.2 - Find a unit vector that has the same direction as...Ch. 12.2 - Find a unit vector that has the same direction as...Ch. 12.2 - Find a unit vector that has the same direction as...Ch. 12.2 - Find the vector that has the same direction as...Ch. 12.2 - What is the angle between the given vector and the...Ch. 12.2 - What is the angle between the given vector and the...Ch. 12.2 - If v lies in the first quadrant and makes an angle...Ch. 12.2 - If a child pulls a sled through the snow on a...Ch. 12.2 - A quarterback throws a football with angle of...Ch. 12.2 - Find the magnitude of the resultant force and the...Ch. 12.2 - Find the magnitude of the resultant force and the...Ch. 12.2 - The magnitude of a velocity vector is called...Ch. 12.2 - A woman walks due west on the deck of a ship at 3...Ch. 12.2 - A crane suspends a 500-lb steel beam horizontally...Ch. 12.2 - A block-and-tackle pulley hoist is suspended in a...Ch. 12.2 - The tension T at each end of a chain has magnitude...Ch. 12.2 - A boatman wants to cross a canal that is 3 km wide...Ch. 12.2 - Three forces act on an object. Two of the forces...Ch. 12.2 - Find the unit vectors that are parallel to the...Ch. 12.2 - a Find the unit vectors that are parallel to the...Ch. 12.2 - If A, B, and C are the vertices of a triangle,...Ch. 12.2 - Let C be the point on the line segment AB that is...Ch. 12.2 - a Draw the vectors a=3,2,b=2,1 and c=7,1. b Show,...Ch. 12.2 - Suppose that a and b are nonzero vectors that are...Ch. 12.2 - If r=x,y,z and r0=x0,y0,z0, describe the set of...Ch. 12.2 - If r=x,y,r1=x1,y1, and r2=x2,y2, describe the set...Ch. 12.2 - Figure 16 gives a geometric demonstration of...Ch. 12.2 - Prove Property 5 of vectors algebraically for the...Ch. 12.2 - Use vectors to prove that the line joining the...Ch. 12.2 - Suppose the three coordinate planes are all...Ch. 12.3 - Which of the following expressions are meaningful?...Ch. 12.3 - 210 Find ab. a=5,2, b=3,4Ch. 12.3 - 210 Find ab. a=1.5,0.4, b=4,6Ch. 12.3 - 210 Find ab. a=6,2,3, b=2,5,1Ch. 12.3 - 210 Find ab. a=4,1,14, b=6,3,8Ch. 12.3 - 210 Find ab. a=p,p,2p, b=2q,q,qCh. 12.3 - 210 Find ab. a=2i+j, b=i-j+kCh. 12.3 - 210 Find ab. a=3i+2j-k, b=4i+5kCh. 12.3 - 210 Find ab. |a|=7, |b|=4, the angle between a and...Ch. 12.3 - Find ab. |a|=80, |b|=50, the angle between a and b...Ch. 12.3 - If u is a unit vector, find uv and uw.Ch. 12.3 - If u is a unit vector, find uv and uw.Ch. 12.3 - a Show that ij=jki=0. b Show that ii=jj=kk=1.Ch. 12.3 - A street vendor sells a hamburgers, b hot dogs,...Ch. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - 2122 Find, correct to the nearest degree, the...Ch. 12.3 - 2122 Find, correct to the nearest degree, the...Ch. 12.3 - 2324 Determine whether the given vectors are...Ch. 12.3 - 2324 Determine whether the given vectors are...Ch. 12.3 - Use vectors to decide whether the triangle with...Ch. 12.3 - Find the values of x such that the angle between...Ch. 12.3 - Find a unit vector that is orthogonal to both i+j...Ch. 12.3 - Find two unit vectors that make an angle of 60...Ch. 12.3 - 2930 Find the acute angle between the lines....Ch. 12.3 - 2930 Find the acute angle between the lines....Ch. 12.3 - 3132 Find the acute angles between the curves at...Ch. 12.3 - 3132 Find the acute angles between the curves at...Ch. 12.3 - 3337 Find the direction cosines and direction...Ch. 12.3 - 3337 Find the direction cosines and direction...Ch. 12.3 - 3337 Find the direction cosines and direction...Ch. 12.3 - 3337 Find the direction cosines and direction...Ch. 12.3 - 3337 Find the direction cosines and direction...Ch. 12.3 - If a vector has direction angles =/4 and =/3, find...Ch. 12.3 - 3944 Find the scalar and vector projections of b...Ch. 12.3 - 3944 Find the scalar and vector projections of b...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Show that the vector orthaab=bprojba is orthogonal...Ch. 12.3 - For the vectors in Exercise 40, find orthab and...Ch. 12.3 - If a=3,0,1, find a vector b such that compab=2.Ch. 12.3 - Suppose that a and b are nonzero vectors. a Under...Ch. 12.3 - Find the work done by a force F=8i6j+9k that moves...Ch. 12.3 - A tow truck drags a stalled car along a road. The...Ch. 12.3 - A sled is pulled along a level path through snow...Ch. 12.3 - A boat sails south with the help of a wind blowing...Ch. 12.3 - Use a scalar projection to show that the distance...Ch. 12.3 - If r=x,y,z,a=a1,a2,a3, and b=b1,b2,b3, show that...Ch. 12.3 - Find the angle between a diagonal of a cube and...Ch. 12.3 - Find the angle between a diagonal of a cube and a...Ch. 12.3 - A molecule of methane, CH4, is structured with the...Ch. 12.3 - If c=|a|b+|b|a, where a, b, and c are all nonzero...Ch. 12.3 - Prove Properties 2, 4, and 5 of the dot product...Ch. 12.3 - Suppose that all sides of a quadrilateral are...Ch. 12.3 - Use Theorem 3 to prove the Cauchy-Schwarz...Ch. 12.3 - The Triangle Inequality for vectors is...Ch. 12.3 - The Parallelogram Law states that...Ch. 12.3 - Show that if u+v and uv are orthogonal, then the...Ch. 12.3 - If is the angle between vectors a and b, show...Ch. 12.4 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - If a=i2k and b=j+k, find ab. Sketch a, b, and ab....Ch. 12.4 - 912 Find the vector, not with determinants, but by...Ch. 12.4 - 912 Find the vector, not with determinants, but by...Ch. 12.4 - 912 Find the vector, not with determinants, but by...Ch. 12.4 - 912 Find the vector, not with determinants, but by...Ch. 12.4 - State whether each expression is meaningful. If...Ch. 12.4 - 1415 Find |uv| and determine whether uv is...Ch. 12.4 - 1415 Find |uv| and determine whether uv is...Ch. 12.4 - The figure shows a vector a in the xy-plane and a...Ch. 12.4 - If a=2,1,3 and b=4,2,1, find ab and ba.Ch. 12.4 - If a=1,0,1,b=2,1,1, and c=0,1,3, show that...Ch. 12.4 - Find two unit vectors orthogonal to both 3,2,1 and...Ch. 12.4 - Find two unit vectors orthogonal to both jk and...Ch. 12.4 - Show that 0a=0=a0 for any vector a in V3.Ch. 12.4 - Show that (ab)b=0 for all vectors a and b in V3.Ch. 12.4 - 2326 Prove the property of cross products Theorem...Ch. 12.4 - 2326 Prove the property of cross products Theorem...Ch. 12.4 - 2326 Prove the property of cross products Theorem...Ch. 12.4 - 2326 Prove the property of cross products Theorem...Ch. 12.4 - Find the area of the parallelogram with vertices...Ch. 12.4 - Find the area of the parallelogram with vertices...Ch. 12.4 - 2932 a Find a nonzero vector orthogonal to the...Ch. 12.4 - 2932 a Find a nonzero vector orthogonal to the...Ch. 12.4 - 2932 a Find a nonzero vector orthogonal to the...Ch. 12.4 - 2932 a Find a nonzero vector orthogonal to the...Ch. 12.4 - 3334 Find the volume of the parallelepiped...Ch. 12.4 - 3334 Find the volume of the parallelepiped...Ch. 12.4 - 3536 Find the volume of the parallelepiped with...Ch. 12.4 - 3536 Find the volume of the parallelepiped with...Ch. 12.4 - Use the scalar triple product to verify that the...Ch. 12.4 - Use the scalar triple product to determine whether...Ch. 12.4 - A bicycle pedal is pushed by a foot with a 60-N...Ch. 12.4 - a A horizontal force of 20 lb is applied to the...Ch. 12.4 - A wrench 30 cm long lies along the positive y-axis...Ch. 12.4 - Let v=5j and let u be a vector with length 3 that...Ch. 12.4 - If ab=3 and ab=1,2,2, find the angle between a and...Ch. 12.4 - a Find all vectors v such that 1,2,1v=3,1,5 b...Ch. 12.4 - a Let P be a point not on the line L that passes...Ch. 12.4 - a Let P be a point not on the plane that passes...Ch. 12.4 - Show that |ab|2=|a|2|b|2(ab)2.Ch. 12.4 - If a+b+c=0, show that ab=bc=caCh. 12.4 - Prove that (ab)(a+b)=2(ab).Ch. 12.4 - Prove Property 6 of cross products, that is,...Ch. 12.4 - Use Exercise 50 to prove that a(bc)+b(ca)+c(ab)=0Ch. 12.4 - Prove that (ab)(cd)=|acbcadbd|Ch. 12.4 - Suppose that a0. a If ab=ac, does it follow that...Ch. 12.4 - If v1, v2, and v3 are noncoplanar vectors, let...Ch. 12.5 - Determine whether each statement is true or false...Ch. 12.5 - 25 Find a vector equation and parametric equations...Ch. 12.5 - 25 Find a vector equation and parametric equations...Ch. 12.5 - 25 Find a vector equation and parametric equations...Ch. 12.5 - 25 Find a vector equation and parametric equations...Ch. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - Is the line through (4,6,1) and (2,0,3) parallel...Ch. 12.5 - Is the line through (2,4,0) and (1,1,1)...Ch. 12.5 - a Find symmetric equations for the line that...Ch. 12.5 - a Find parametric equations for the line through...Ch. 12.5 - Find a vector equation for the line segment from...Ch. 12.5 - Find parametric equations for the line segment...Ch. 12.5 - 1922 Determine whether the lines L1 and L2 are...Ch. 12.5 - 1922 Determine whether the lines L1 and L2 are...Ch. 12.5 - 1922 Determine whether the lines L1 and L2 are...Ch. 12.5 - 1922 Determine whether the lines L1 and L2 are...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 4144 Use intercepts to help sketch the plane....Ch. 12.5 - 4144 Use intercepts to help sketch the plane....Ch. 12.5 - 4144 Use intercepts to help sketch the plane....Ch. 12.5 - 4144 Use intercepts to help sketch the plane....Ch. 12.5 - 4547 Find the point at which the line intersects...Ch. 12.5 - 4547 Find the point at which the line intersects...Ch. 12.5 - 4547 Find the point at which the line intersects...Ch. 12.5 - Where does the line through (3,1,0) and (1,5,6)...Ch. 12.5 - Find direction numbers for the line of...Ch. 12.5 - Find the cosine of the angle between the planes...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5758 a Find parametric equations for the line of...Ch. 12.5 - 5758 a Find parametric equations for the line of...Ch. 12.5 - 5960 Find symmetric equations for the line of...Ch. 12.5 - 5960 Find symmetric equations for the line of...Ch. 12.5 - Find an equation for the plane consisting of all...Ch. 12.5 - Find an equation for the plane consisting of all...Ch. 12.5 - Find an equation of the plane with x-intercept a,...Ch. 12.5 - a Find the point at which the given lines...Ch. 12.5 - Find parametric equations for the line through the...Ch. 12.5 - Find parametric equations for the line through the...Ch. 12.5 - Which of the following four planes are parallel?...Ch. 12.5 - Which of the following four lines are parallel?...Ch. 12.5 - 6970 Use the formula in Exercise 12.4.45 to find...Ch. 12.5 - 6970 Use the formula in Exercise 12.4.45 to find...Ch. 12.5 - 7172 Find the distance from the point to the given...Ch. 12.5 - 7172 Find the distance from the point to the given...Ch. 12.5 - 7374 Find the distance between the given parallel...Ch. 12.5 - 7374 Find the distance between the given parallel...Ch. 12.5 - Show that the distance between the parallel planes...Ch. 12.5 - Find equations of the planes that are parallel to...Ch. 12.5 - Show that the lines with symmetric equations x=y=z...Ch. 12.5 - Find the distance between the skew lines with...Ch. 12.5 - Let L1 be the line through the origin and the...Ch. 12.5 - Let L1 be the line through the points (1,2,6) and...Ch. 12.5 - Two tanks are participating in a battle...Ch. 12.5 - Give a geometric description of each family of...Ch. 12.5 - If a, b, and c are not all 0, show that the...Ch. 12.6 - a What does the equation y=x2 represent as a curve...Ch. 12.6 - a Sketch the graph of y=ex as a curve in 2. b...Ch. 12.6 - 38 Describe and sketch the surface. x2+z2=1Ch. 12.6 - 38 Describe and sketch the surface. 4x2+y2=4Ch. 12.6 - 38 Describe and sketch the surface. z=1y2Ch. 12.6 - 38 Describe and sketch the surface. y=z2Ch. 12.6 - 38 Describe and sketch the surface. xy=1Ch. 12.6 - 38 Describe and sketch the surface. z=sinyCh. 12.6 - a Find and identify the traces of the quadric...Ch. 12.6 - a Find and identify the traces of the quadric...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2930 Sketch and identify a quadric surface that...Ch. 12.6 - 2930 Sketch and identify a quadric surface that...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - 3942 Use a computer with three-dimensional...Ch. 12.6 - 3942 Use a computer with three-dimensional...Ch. 12.6 - 3942 Use a computer with three-dimensional...Ch. 12.6 - 3942 Use a computer with three-dimensional...Ch. 12.6 - Sketch the region bounded by the surfaces z=x2+y2...Ch. 12.6 - Sketch the region bounded by the paraboloids...Ch. 12.6 - Find an equation for the surface obtained by...Ch. 12.6 - Find an equation for the surface obtained by...Ch. 12.6 - Find an equation for the surface consisting of all...Ch. 12.6 - Find an equation for the surface consisting of all...Ch. 12.6 - Traditionally, the earths surface has been modeled...Ch. 12.6 - A cooling tower for a nuclear reactor is to be...Ch. 12.6 - Show that if the point (a,b,c) lies on the...Ch. 12.6 - Show that the curve of intersection of the...Ch. 12.6 - Graph the surfaces z=x2+y2 and z=1y2 on a common...Ch. 12.R - What is the difference between a vector and a...Ch. 12.R - How do you add two vectors geometrically? How do...Ch. 12.R - If a is a vector and c is a scalar, how is ca...Ch. 12.R - How do you find the vector from one point to...Ch. 12.R - How do you find the dot product ab of two vectors...Ch. 12.R - How are dot products useful?Ch. 12.R - Write expressions for the scalar and vector...Ch. 12.R - How do you find the cross product ab of two...Ch. 12.R - How are cross products useful?Ch. 12.R - a How do you find the area of the parallelogram...Ch. 12.R - How do you find a vector perpendicular to a plane?Ch. 12.R - How do you find the angle between two intersecting...Ch. 12.R - Write a vector equation, parametric equations, and...Ch. 12.R - Write a vector equation and a scalar equation for...Ch. 12.R - a How do you tell if two vectors are parallel? b...Ch. 12.R - a Describe a method for determining whether three...Ch. 12.R - a How do you find the distance from a point to a...Ch. 12.R - What are the traces of a surface? How do you find...Ch. 12.R - Write equations in standard form of the six types...Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - Determine whether the statement is true or false....Ch. 12.R - a Find an equation of the sphere that passes...Ch. 12.R - Copy the vectors in the figure and use them to...Ch. 12.R - If u and v are the vectors shown in the figure,...Ch. 12.R - Calculate the given quantity if...Ch. 12.R - Find the values of x such that the vectors 3,2,x...Ch. 12.R - Find two unit vectors that are orthogonal to both...Ch. 12.R - Suppose that u(vw)=2. Find a (uv)w b u(wv) c v(uw)...Ch. 12.R - Show that if a, b, and c are in V3, then...Ch. 12.R - Find the acute angle between two diagonals of a...Ch. 12.R - Given the points...Ch. 12.R - a Find a vector perpendicular to the plane through...Ch. 12.R - A constant force F=3i+5j+10k moves an object along...Ch. 12.R - A boat is pulled onto shore using two ropes, as...Ch. 12.R - Find the magnitude of the torque about P if a 50-N...Ch. 12.R - 1517 Find parametric equations for the line. The...Ch. 12.R - 1517 Find parametric equations for the line. The...Ch. 12.R - 1517 Find parametric equations for the line. The...Ch. 12.R - 1820 Find an equation of the plane. The plane...Ch. 12.R - 1820 Find an equation of the plane. The plane...Ch. 12.R - 1820 Find an equation of the plane. The plane...Ch. 12.R - Find the point in which the line with parametric...Ch. 12.R - Find the distance from the origin to the line...Ch. 12.R - Determine whether the lines given by the symmetric...Ch. 12.R - a Show that the planes x+yz=1 and 2x3y+4z=5 are...Ch. 12.R - Find an equation of the plane through the fine of...Ch. 12.R - a Find an equation of the plane that passes...Ch. 12.R - Find the distance between the planes 3x+y4z=2 and...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - 28-36. Identify and sketch the graph of each...Ch. 12.R - An ellipsoid is created by rotating the ellipse...Ch. 12.R - A surface consists of all points P such that the...Ch. 12.P - Each edge of a cubical box has length 1 m. The box...Ch. 12.P - Let B be a solid box with length L, width W, and...Ch. 12.P - Let L be the line of intersection of the planes...Ch. 12.P - A plane is capable of flying at a speed of 180...Ch. 12.P - Suppose v1 and v2 are vectors with,...Ch. 12.P - Find an equation of the largest sphere that passes...Ch. 12.P - Suppose a block of mass m is placed on an inclined...Ch. 12.P - A solid has the following properties. When...

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