Each edge of a cubical box has length 1 m. The box contains nine spherical balls with the same radius r. The center of one ball is at the center of the cube and it touches the other eight balls. Each of the other eight balls touches three sides of the box. Thus the balls are tightly packed in the box (see the figure). Find r. (If you have trouble with this problem, read about the problem-solving strategy entitled Use Analogy on page 98.)

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Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621
BuyFind

Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621

Solutions

Chapter 12.P, Problem 1P
Textbook Problem

Each edge of a cubical box has length 1 m. The box contains nine spherical balls with the same radius r. The center of one ball is at the center of the cube and it touches the other eight balls. Each of the other eight balls touches three sides of the box. Thus the balls are tightly packed in the box (see the figure). Find r. (If you have trouble with this problem, read about the problem-solving strategy entitled Use Analogy on page 98.)

Chapter 12.P, Problem 1P, Each edge of a cubical box has length 1 m. The box contains nine spherical balls with the same

Expert Solution
To determine

To find:

The radius r

Answer to Problem 1P

Solution:

The radius of each ball is

r =3-32m

Explanation of Solution

1) Concept:

We create an analogous problem in two dimensions and use the strategy in two dimension to solve the problem in three dimension.

2) Calculation:

Since three dimensional situations are often diffucult to visualize and work with, we create an analogous problem in two dimensions and use the strategy in two dimension to solve the problem in three dimension.

Let us try to find the analogous problem in two dimensions. The analogue of a cube is a square, and the analogue of a sphere is a circle.Thus, a simillar problem in two dimensions is as follows:

What is r, if five circles with the same radius r are contained in a square of side 1m so that the circles touch each other and four of the circles touch two sides of square, as shown in the figure,

Calculus (MindTap Course List), Chapter 12.P, Problem 1P , additional homework tip  1

By pythagorean Theorem, the diagonal of the square is 2

Also from the figure, the diagonal of the square is 4r+2x

Therefore, 4r+2x=2

But, x is the diagonal of the smallar square of side r.

Therefore,

 x=2 r

Therefore,

4r+2x=2 implies that

4r+22 r =2

That is, 4+22 r =2

Therefore,

 r =24+22

Let’s use this idea to solve the original three dimensional problem.

Calculus (MindTap Course List), Chapter 12.P, Problem 1P , additional homework tip  2

The diagonal of the cube is 12+12+12=3

And the diagonal of the cube is also 4r+2x.

Therefore, 4r+2x=3

 where x is the diagonal of the cube with edge r.

Therefore,

x=r2+r2+r2=3 r

Therefore,

4r+2x=3 implies that  4r+23 r=3

That is, 4+2 3 r =3

Therefore,

 r =34+23

By rationalization,

r =23-32

Therefore, the radius of each ball is

r =3-32m

Conclusion:

The radius of each ball is

r =3-32m

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Chapter 12 Solutions

Calculus (MindTap Course List)
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. 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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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